Information processing in molecular systems

INFORMATION PROCESSING IN MOLECULAR SYSTEMS

1. lntmduction

Much of “UTintuition about information processderives from experience with classical machines such aa computers. These are really a very restricted class of information processing devices. In this paper I will showthatmany of the concepts derived from classicrdm;ahi”es are inappliaddo to molecular sy* terns and that such systmns have essentially different ca~bililier and limitations. Jn particular this is true for the ~pmcessen of heredity and self-reproduction in biological systems. Chemical systems process iaformatio” in what might be called a hierarchical mode, as opposed to the &gle level mode. characteristic of present day computers or propmed constmctors. The objects(enzymes) which determine the behavior of biochemical systems are desaibed genetically at the primary level of stmctme, but function at tie tertiwy level, on the basis of specificity inherent inihcir three dimensional ing

shape and charge distribution. This sharply distinguishes molecular systems from conventional corn. puters where the objects which determine behavior are always described at the sane level at which they function. The hierarchical nature of biological mdecules is well know”. However, the significance of m&cular hierarchy far Infomlation procesing, its ultimate capabilities and iimitations, is not well understood. My main aim in this paper is to investigate this by showi”B precisely where the analogy between dassical systems and their chemical counte~pats is broken. I will do this by chamcterizing Information proceaing and its relation to self-reproduction in general. T%enI will neview the assumptions which underlie information processing and selfrepredaxio” in classical ma chines, macihes for which WI theories and intuitions are well developed. Finally I will consider how these asmnpnonr m”st be weakened in systems of molecular dimensions. This has important bnplications for information prwessiag in such systems, especially for their pate of operation PIid the rate at which they can evolve. In partialar, the hierarchicpl

mode allows for “adialratic” modification of the behavior o? the swtem. lherebv facilitatine evolution processes.How&, it’i!; incompatible v& the auto. mata theory notion o!’ effective computation and therefore with concepts of programming as a~rently understood.

2. lnfommtion pmcess~ 8s selective dis3ipation Biological nature is open to energy, primarily energy from the sun. This enters the biosphere with a certain rmge o-f frequencies and leaves with a louw range of frequersies, i.e., us hat. From the standpoint of themmdynzuuics there are many alternative pathways which may mediate thir dissipation. These &ernatives form the basis of information processing (or selectivedtssipatto”) in biology. The dissiuativa flow of enerav drives the cyclic tnnsfonatibns of matter in the biosphere. In~thts cass the alternatives are provided by the alternative cycles or sequencer of reactions. For example, coosider fig. I. This is a chemical system operating be. tween ;wo temperatures. The flow of energy through this system results in an increase in entropy and the&we in loss 0” information about the nwt state of Ic world. This might be mediated by &ie BCD

or cycle AEF of through other reactions (broken lines). The rate of entropy production, or the dissipa. tion, can be incnxsed by adding catalysts to the system. These can increase the dissipation in altemative ways, depending on which reactions they speed up. If the cataly!;ts (Eij) increase the rates of reactions in cycle BCD this will provide the main mechanism of dissipation. In generJ the actual pathway of dissipatioo is determined by the cnialysts since these determine the rates of the various transformations. Essentially the catalysts restrict the dynamic motions of the system in its phase space. This is clear if some sequenceof reactions branching off the BCD cycle is driven to completion. In this case the possibility of dissipation tbrou& the AEF branch of the reaction scheme may be entirely precluded, i.e., certain regions of the phase space may become completely inaccessible. In general, information processing(as opposed to simple transmission) requires selective loss of information about the past. This is a precondition for many-to-one operations such as resetting as well as for logical operations such as disjunction. Reaction schemessuch as the one discussedabove provide the predominant mode of information pm cessingin biology. However, selective dissipation can also occur without chemical change or exchange of

processes

matter. This is true in ordinaly computing machines. Here information processing is based on the altemative pathways by which electrical or mechanical eoergy can be conwted to heat, as in the device illustrated in fig. 2. Clearly such “classi&’ information processing devices are special cases as compared to open systems or systems undergoing chemical trans. formation.

3. Selective diiipglion

and self-reproduction

Most pathways of dissipation are quite uninteresting. Only a minority are associated with behavior which Z signilicant from the biological point of view. The pathways of most prominent interest are those associated with the processes of self-reproduction and evolution. In general, dissipation which really subserves an information processing function has an essential relation to these processes. Information processing devices, like aU machines. tend to break down. In order for such devices to persist indefinitely-to have an indefinite half-lifethey must be able to repair themselves or must be reconstructed by a largzr system which is self-repairing.. Naturally the repair apparatus is also subject to breakdown and mmt itself be repaired. If thii is not to lead to logical difticullies some general repair ap pamtus must act on a number of others. The simplest case would be that in which one universeI repair process operated ofi all others, including itself. In fact self-reproduction is just such a process. Here one base set of Constraints is sufficient to effect the repair i1o matter what is wrong 1.1 detail. Futhemwe, selfreproduction implies the repair of these base constraints themselves. Self-reproduction is the limit of

self-repair. Indeed it is a universal mechaniim of selfrepair. Naturally, the proce~ of self-reprodoction is sub. ject to error, but if asufticientnumberofoffspringare produced one of these will be able to repeat the process, assuming that the new envimnment is sufficient. ly similar to the parent’s environment In fact Lafgren (19f.3) has formally shown that a self-repairing syr tern with an unbounded lifetime must incorporate soms growth scheme such as dfireprodoction. I! is obvious that oresent day biolopical wstems can produce offspring with complexity equal to or greater than their own. Van Neumann (1951) has given a general construction for an abstract system with this capability. The flow diagram of this constructicn is illustrated in fig. 3. The system has a description (genotype), a construction szvice which maps the description into the system dextibed (or phenatypc), P replicating device which transcribes the description, and a control device which supervises the sequence of opratjons. The requirement for double function (construction and replication) follows from the fact thnt the description must not describe itself since would then have to describe the description of italf, and so forth, leading to an intinitc regrr?ss ‘(Moon:, 1964). Self-repmduction takes place when the description is read by a construction device wb&h is itself described by the description: this is the functional relation which is the sine qua non of self-

L

reprCiiUCti"".

The requirement of double function is predicated on the assumption of a distinct description. Later it will be clear-that the presence of a histinet, static description simplifies the dynamics of self-reproduelion (cf. Arbib, 1969). The des+tion also subsemen the repair function of self-repr~~doction since it pro-

vides the system witi minformation stow which tai!ors its constraints. Tbis ““ens the oossibilitv for evelutionary modification. In’order td play this role the c’escriptionmust be enew degenerate, i.e., its physical properties must be independent of what it d* scribes. If this were nlot so, certain descriptions would be altogether more i,mprobable than others (the description would not have a high information capacity) and the pcaRilities for evolution would be limited. In fact computer experiments have shown that bias. ing the probabilities of different description:;(by biase the mutation probabilities) retards the rate ofevolutio” (Conrad. IY69). From the strictly logical point of view there is a certain freedom bt the overall flow diagram of selfreproduction, i.e., the arrow in fig. 3 can be rem. ranged somewhat For example, the constructor may read from the parent description, the daughter d6 scription. or some combination of these. In particular there is no logical reason why the ConStmctoIcannot, uwiify the description or eve” describe itself (cf. My.dll, 1964; Thatcher, 1%2). However, much of this arbitrariness is removed by the repair role of selfreproduction. From this standpoint there must be some ratchet which isolates the descriution from the e~virofunent, so thatA controk the e&alInent bUt not conversely.If this were not so the naturaldisipa. tive tendenciesof the phenotypewould be communicated to the Lxrlption, therebynegatingits valueas an iaformatiou store. This isolation of the description is known 15 the strong principle of inheritance or Darwinian property. There is evideot zuwdogybetween the flow diagram I” fig. 3 an: the operation of biological systems. As is well know”. the dwble wdlng aspect of self.reprP duction (trauscription md translation of the dewrip tion) io P prlmuy feature of seJf.reproductioa at the cellular level. The g’aetic dwiption (DNA) is highly cneru degenerate aad a8 far w is known biological systems obey the strong principle of inheritance (e.g., central dogma). However,the analogy must be viewed with care as long as dercriptioc, tranrlation, ami con. stmction retain their cust&ry sigoiflcance in terms of &s&al aY*tema.In fact the UhYsiCaIsienaxlxe of these pIo&es change3f&al&- an * flkction of rate. In nhat follow I will clarify this hy studying 5uch prowsea in oidinaty computen aiwe these are weII underrrocd. Then we can see bow such systems

differ from those in which individual molecules play a prominent role. 4:Self-leproduction in classicalsystems

The computing machines which we know at the present time “re made up of a fixed set of macroscop ic components. These machines manipulate symbols, ordinarily represented as states of components. The computing muchine is a closed system as far as components are concerned. Thus the manipulation of symbols in computatation (involving weak interactions or bonds) is sharply distinguished from the manipulation of components in construction (iivolving strong interactions or bonds). Needless to say a computing machine which reproduces itself must be able to manipulate components. For example, imagine a pool of parts with an exteiual enelgy source in which some pattern of activity (system) branches to form two daughter patterns. We say that self.rep:oduction takes place if each of the daughter patterns is essentially similar to the parent. No machine with this capability has ever been built but programs (or constructive~roofs for the existence of such systems) have hen worked out in detail (van Iieumann, 1966). In order to destgu such a system it must be possible to prescribe just how the system will manipulate each part in the pool. This is not the same PI predlet. ing the behavior of the system on the bats of interactions amoog the parts-in general this involves the solution of very complicated equations of motion. Furthemwre, it begs the question of design since we wouldn’t know how to set up these equations unless we had a clear idea (on a prescriptive basis) that they descdbe a sclf-reproduclng system. I” Whet fdlOWSI will ~&UPCthe physicalpropertieswhich a system must have if it is to exeoute prercribedbehavior. These are the propertieswhich underliethe idea of effectiveness,i.e., which enablethe systemto execute al&&thmiiproceuer. 4.2. Comprters J will begin by exanining the physicalproptzties of computersv/h&hunderlieeffectivecomp~tetion. l&hen it will be poaalbleto undentandthe cwaterpart

properties for construction. A digital computer is sometimes described as a finite automaton along with a memory space which it CP” maimdate. An aut0maton is a system with n fmite set-of inputs(X), a finite set of internal states (0). .-,, and a finite set of outputs(Y), along with transition functions b : X(r) X Q(t) -I Q(t + 7) (next state function) w: X(t) X Q(t) -f Y(tt7)

(next output Cmctio”)

where t 6; the time and r is some time interval. These are tables which determine the internal state and output for the next interval of time give” the present internal state and inout (illustral.rd in tie. 2). The *Me of a computer isits configutatio” of parts, i.e., it is give” by the states of its components and their i”terco”“ectio”s. A classical automaton has the following properties: Al. It is a” elementety ph$sical device (unit) which executes Bdefinite function, or AL It is a collectionof elementarv deviceswhose inputs nnd outputs am linked. _ AXThefunctions executed by the units do not change tie” thr.y are linked together. Therefore the composite device also executes a definite function, deriviblq from the functions executed by the individual ““its. Property A3 dependls on the fart that the inputs received by a unit from other units correspond to (or have bee” coded into) members ofits input set. Notice that the tinction executed by the device differs from the function (or equation) which describes its behavior from the standpoint of physics. The function executed by the automatOn is wdfied by transition tables which ordinarily have mani fewer entrier,ihan external conditions and physical states of the system. As :S clear from fig. 2 the systems which realize such tables have equivalence classes of histo. ries, i.e., many rtatw and inputs a.ttime t ca” give rise to the rsmc state (or output) at time f+r. This is not (111~for ordinaty mechanical systenx who= past and futue are completely determined by their state eve, B small interval of time. In, order for such systems to approximate zmtomatathey must have classes of states which are dqoivalmt from the macmscop1c point of tiew. 1

I

Properties Al -A3 have the following physical EOIA rel&S.: PI The behavior of the system is dominated by CODstraint. Forces of constraint restrict the dynamical motions of the system to certain regions ofits phase space.These reflons correspond to states of the automaton, i.e., are cqtdvalence classes of physical states. Therefore the initial canditio”~ are also split into equivalence classes,corresponding to the inputs to the automaton. P2.When onifs are linked the possibIe inputs to at least one of them are restricted. Thii means that the initial conditions of linked systems are not arbitrary. I3 The way jl which the dynamical motions of the system arc restricted by the forces of constraint is not modified by linking systemstogether. Property PI, the nonhdonomic constraint aspect of machine behavior, has been emphasized by Pattee (1966). This propsrty means that the unit8 are s&ctixly dissipative, or exhibit relective 10~ of infonnatian about the part. In the ~846oflelementsry units the maximum number of equivalence claws of initial conditions correspond to the number of inpots. I” general the actual number of inputs which ca” he distinguished depends MI the number of time ptiodr which have elapsed (unless the system simply cbnts). Thus information may also be destroyed through the state to state transitions of the unit (computation). The selectitity of this type of dissipation is increased by lting units, where this just mea”s that some units are constrained to act on restricted sets of initial conditions (property P2). In general, initial conditions are just those facets of the physical world for which them is no theory (Pattee, 1%8), but this is not the case withi” P linked automaton. The main dgniticance of property Pl is that there is 2 definite relation between the stmctwe and functlo” of the units. Once this relation ia established the equations of motion do not have to be consulted in order to determine the functtonelly si@icant be havior. Accoxding to property F’.3this structurefunction relation is not disturbed by linking units; interconn~ctio” of units only revtricts, and “ever enlarges, the set of possible input seqwncn. ‘i&ismeans that ““its either do not respondto other unitsin the coilection of reyond to deBned outputs of these units. fhey never respond to, or interactwith, the

collections

global properties of of other units. The absence of globalinteraction makes it possible to embody the precise conditions ggyeming the behavior of the computer in a symbolic stroctwe. I will say that such e system acts in a one level mode. This is because symbolic stroct”~es have no global proper. ties. q., do not fold like polypeptide bhains. (Of COUN: in enothel sense computers mey be described l&Is-at the level of electronic circuitry, in terms of the language of the machine, et various higher layers of language which a,e constructed from the language of the machine, and so forth.) These properties, eqxcially the single level property. ere evident in any forn~alizaticn nf algorithmic processes. These always involve symbol by symbol menipulstio”. This is necessery because each symbol represents tba state of some unit. eve” if this is only a memory unit. If sequences of symbols are trans. fo”“ed in a single step this txansfommtion most be nonselectiw, i.e., independent of the sequence. If this were not so the transition function governing the trenrfommtion would not be defined since property A3 (or P3) would, in general, fail. The sin$e level property is clear, for example, when al~orithndc pro. fesses a”! expressed by the base functions and schemat” of recursive functions. The base functions allow symbol by symbol manipulation of sequences of symbols: for example, selection of the first symbol, deletion of the first symbol, concatenation of symbeds. The schemata e&v for Ii”!@ of the base f;“c(ions, for exan!ple by composition or recursion. There ere e number of theoretical models of cornpoters, all of which ere capable of computing the stwe class of fonctions (partial recursive functions). The best known of these ma&i”es is the Twine me. chhxe, which is dmply n finite aotomaton elon;with a memory tape which it can na.d, write on, erase, and “IOVC. This reads one input symbol et e time, rewriter thi: input symbol according to its present state. moves the tqee, and goes or. to a new state. The transition functions (program) ere communicated to the Turb~g macchinc by selecting the appropriate elementary wits (which pelfon” known functions) end lha”g their. in the appropriate way. This process can itself be expressed algorithmically. e.g., appropriate dementary units may be selected and linked in such e

atmany

way that they cull accept the description of any spe. cisl Twi”g machine es input and compute the seme

functid es this special machine. Such e machine is un’versel. Any general purpose computer is universal in this sense. Accordine to Twine’s hvootbesis a universal machine can be ~ogramr.~ed in i&h e way that it can compote any effectively computable function, or any function computable bysome recipe. Now it is clear why computers are effectively programmable, i.e., why algorithnx can be communi. cated to them: 1) We orovide e functional descriotion f~rescnotion\ ‘_ . for ;Ite process to be perform&. This functional prescription is implemented by translating it into the stmctw of some part of the

2)

’

COFIlpUt~L

3) This translation process is automatic because the stmct”re(and therefore the transition function) of ““its is not modified by linking them to other units. In short, computers are effectively programmable because the trans:tion function which governs their behavior is explicitly represented in their circuitry. 4.3. conrhucton Now we can consider the physical assumptions which underlie whet mieht be celled effective constmctio”, i.e., constmc& according to a definite procedure. Von Neumann demonstrated the possibility that systems with the double reading properly (fig. 3) could be universal in two senses. First they could be universal computers. Second they could be universal ir the sense that they could constmct a”y system for which a description is provided. If ouch systems are provided with their own description they are selfreproducing. If they ere provided with a description of a more complicated system they will produce more complicated offspring. I shell not describe the operations of these systems in detail (for the simplest, but still involved constmclion. see Arbib. 1967). Whet is imoonant. from the phy&l point of view; is that the &its mist be able to sense patterns in their environment and transform these in a prescribed manner The system must esteb. lish a certain coordinate system in ihe environment and move parts in ways which are independent of the precise (or reel physical) details of the parts on this coordinate system. Essentially the dynamical motions of the SystLn nut be constrained so that it selective-

ly loses certain mformation about its enwronment. II this were not so, if certain details were not ignarable, the system could not perform these recognition and transformation operations m a finite numner of steps. This whole problem of pattern recognition and transformation is simplified if there are external constraints which restrict the patterns-for exxnple, ti the coordinate system IS a real external constraint. In general this is the case-the exntence of self-reproducing systems is ordinarily proved with tessellation automata. Essentially the tessellation is a grid each block of which is a finite automaton capable of assuming a number of states and of receiving inputs from neighboring automata. Preciw self.reproduction Fakes piace when some grid pattern generate*; another with which it could be interchanged without any detectiblc change in the appearance of the grid. This Fonstmint ensures that each of the units will per%rr.l definite functions, i.e., they ~upprev the necessiry of consuiting the equations of motiun to determine their functionally significant behavior. This is obvious in the case of tessellation automata-here we really have the trivia: case where the grid suppresses the dynamiccl properties of the blocks altugether. Thus the efiectivcness property is porsible for construction ii the manipulation of compunents is suhject to constraints in such a way that it can be corn. pletely controlled by an effective computation process. Thii is a precondition ior programming the constnwtor. Self-reproducing systems can exist without this property-we will discuss ruch systems shortlybut we cannot communicate programs IO them. The program itself is provided by the description. This is true even if the description is :1 blueprint in the conventionaisense, i.e., a structural description of

the system to be built. The biuepriat is a program because !t directs the action of the constructor-it is a functs”“?! prescription. for the behsvior of the constructor under the construction rules embodied therein. it is possible to detenuinr the formation of the daughte, system from the description and the construclion rules without consulting the laws of physics. In tius sectwn I have emphasized the assumptions which underlie the most elementary facts about computers and constructors. The main point is tlmt a system executes &oritbmic processes if and only if we can describe these in terms of interconnections of units to which definite transition functions ere assigned. This IS not true ior all physical processes-not every physical (or mechanical) process is an “lgorithmic rroce~. A system can execute algorithmic processes only if it operates in a single level mode. Thea Inode of operation underlies cifectivc computa. lion, as we have seen. It also underlies effective con. struction since this is based on effective wmputa. tioa-essentislly we must assume that modification of the artangemcnt of units does not affect the functions which they execute.

5. Self-reproduction

in chemical systems

As is well known reproduction in biological systems is mediated by the specific (or selective) cross Icatalysis of nucleic acids add Proteins (fig. 4). The description is in large measure provided by DNA. This is read twice, once for irimscription to new DNA and on2 for translation, mainly to primary structure of protein. In present day cells this cross catalysis is o.ulte esymmetrical with DNA really playing the role

of au information store which tailors reaction constraint’r.As previously discussed,the DNA description EI~RS~nformntionwell because it possessesthe enerw bgeoeracy property-its tertiary properties are quite independent of its primary structure. Thus the specificity of I.ranslation is determined by the conform

Ination of protein enzymes, RSwIl asvarious adaptor molecules and molecular aggregateswhxh function as aclive geomeoical constraints. Apparently the speciBcity for transcription is also in the proteins (Commoner, 1964). However, the sequence of transcribed as well as trawlsted DNA is ordinaBy determined, spart from mutation, by the sequence of parent DNA. The main difference between biological systems and classical machines arises from the fact that trans. lation is only an aspect of construction (see fig. 5). In biological systems translation is only a coding process which saws to break the degenerzcy of the gene. type. This means that construction Is not pr~crtptive. It is an energy proces_-erscntIaUya constrained minimIzatIon of free energy. Naturally energy Is b:quired by classical computers but this Is irrelevant to the transition tables they realize. The I& of these energy proo:sses spans *he fold. tog of tkz protein moiecule, the interaction of dIfferent mdecolcs sccordtng to the spenficity inherent

in the three dbne8siond &ape and charge distribution assumed in these folding processes (self.asscmbly), as well as the specific interaction ofenzyme and substrate. These spec~c interactions determine the rate constants of cellular proce%er,*oor more precisely, determine to what regions of its phase space the system is redrtctcd. In short they determtne the behavior of the system orhow it proc&es information. To some extent these processegare influenced by external, pre-existing structures. Tertiary structure nrises from the energy of interaction of mmy amino acids. but of course it is possible that it is also in. tluenced by tanglinq as the protein comes off the ribasome. Interactions between different proteins to form quaternary stmcture or between ewymes and substrates may be constrained by pre-existing cyte logical stroctwe. There is increasing evidence thal such stmcturc plays an important role in determining the rates of cellular processes; but it does not sup press the dynamics of the system in any way comparable to a tessellation. It is important that such specific interactions are restricted to a crucial range of sizes and are not possible in classical computers. The components must be large enough so that weak interactions (relative to kT) can have a stroog additive effect. On the other hand the components must be small enougJ~so that they can diffuse and interact. In general diffusion coefficients fatI off rapidly with ma wheress weak interactions increase with the number of possible points of contact and therefore roughly wi$ mass. It is these interactions-hydrogen bonding. van der Wa;ll’sforces charge fluctuation interactions, and so forth-which give the mdecule its specificity. Natorally tbe optimum size for such specific interacttonl depends on many factzs-the geometry of the mole. cule, the types of forces and whether or not they saturate (as do dispersion forces), the time scale on which the interactions must proceed, as well as the geometrical constraints which influence these rates. Hcwever it is clear that there is a definite range of sties above and below which we cannot expect interactions based on folding processesto play a practical d?.

These facts are the basis of Important differences between classical and chemical svstems. In bidow the DNA provides a primary &ctuml descriptioi, not a prescriptive or functional description as does

the program of a classica computer. In particulai biological molecules interact with each other on the bask of their global properties-they may operate on smgle constituents, but they dt3 not interact on a const~tu. ent by constituent basis. liuwever, they are described

wsal Turing machine can be represented ly by a sequence of the fom

(genetically) on such a basis. This is why biological systems process information in a hierarchical mode, i.e., in a made which depends on an admixture o$ strong and WC& bonds. Such systems cannot be programmed in the conventional sense because the genetic description just constrains certain ioteractiofs. hut not in a way which eliminates the need to consult physical laws. Notice that the importance of minimization of free energy in molecular systems precluCes a simple reverse reaction wixch would allow changes m primary structure of protein to be encoded in energy deg:nerate DNA. The strong pnnciple of inheritance is based on this energy cmsideration in molecular

whl:re 4, and qf are rhe inhal and final ~,tatcs, xi and yk i’re the mpdr end o’z!put. X, C (mtegersj, X, n X, = B for u $ u. and the state of the uth unit in the sequence is accessed when t=t0+Xy7. This accessing IE eelf cantlolled by the program under the direction of ihe mterpreter program which gives the machine its miversal property. The temporal sequence of the inpI states determines Ihe state trasitions and the input-output behavior of the automaton becnux the tsar sition f!unctions of units and their composition are definite (properties Al-A3). ,uow suppose that XI =X2. In this case qi nnd xi are

constrained

systems as well as on the reliahilily viously discussed.

6. Hisrarchicnl

consideration

In this section 1 want to consider how assumption Al-A3 (Pl-F3) are modified in chemical systems. These assumptions can be represented in the folIonlug formal way. Once we have a transition table for a finite automaton (sect .4.2) we can determine the State transitions under sequences of inputs. me., 8 k?u(t).~,,(01

= q&f +r)

and

=4&+r)

* 6 [q,(r),x,(rlx,(r+

?)I = q&t

27)

where the qi are members of the state set and the xi are members of the input set. On this basis a multiplication table can be built up for the inputs. It can represent the state to state transitions, as above, or it can be constructed to represent the inputGoutput he. havior (the w map). The essential property of a compuwr (which die. tinguishes it from B fmite automaton) is that the time order of the inouts for the acccssibilitv of sta!es in

.

memory) transition

is under system control. functions (data program)

.

acwssed

at the same time, i.e., we have

pre-

mode of computation

sk,wJ,(01

symbolical-

For en?mp:e, the provided to a uni.

whw the dot rejxesents no arbitrary state. The state transition is underivable from the state transitions for q, and x, in seqi~ence. They now form a new state, associated with i; new, larger unit. In genenl if WC increase the allowable equalities among the h,, we increase the number of possible input states. The effect of these de nova states on the behavior of the system cannot he determined from their fmer lewl of structure without reference to physical reasoning or expriiment. Naturally wi: may have strong physical jr arralcgical reasons for anticipating these effects, but such reasoning is not a featwe of algorithmic infor. mation processing. We have seen. from the orevioos section. that this is the utuatioil in the cell. Here the fine level of stmctore 1s the sequence of amino acids in protein-the function uf tie protein is determined by the energy of interachon of many amino acids simultaneously, not Just linear nearest neighbors. As a consequence the DNA, which codes for this fmer level of structure, is entirely disanalogous to a conventional computer pr~.gram. It is dieanalogous because the effect of the DNA sequence on the cellular apparatus is not built up from the sequential action of its manipulable elements, i.e., it is not possible to multiply the states of these units as if they were temporally distinct, o* independently accessed inputs.

The non~sequential character of the description elises from the inappbcability of assumptionA3 (Q3). The transitior function of the cell cannot be derived from transition functions of its manipulable units and the way they are linked together. et least without constdtina eouations of motion. This is the reason for the inter&& within blocks of manipulable “nits in tht!, descriplion: the description is accessed globally (rather than on a constituent bjl constituent basis) r:ter it is translated to primary stmct”re of protein. Thh;rfact cannot be ignored because it is the manip”. lation of these “nits which modifies the systaxn. ‘Thefahre of A3 has three co”seque:tces.The first is a techrical one and is not too important. If X,= A, for an arbitrary number of values of u, this means ti,t there ,ue an rrbitrary number of possible &men. 4aty inwt 3 to the automaton. Thereiore it is not a finite automaton (assumption A!). This is the case in the ccl! since arbitrary cha”ges it? the primary sttucturr of DNA are admisssble. Akrnatlwlv each novel

DNA sequence tray bc viewed as belo”& :o a different state of 2~: r~sten; Szn there is no fmite set of states. Ms’%ircr,the biologal cell mi& ail! be ;pproximated by P” elementely finite automaton, with P deftite number of states and environmental inputs, as lung as ihe variability of the Sr?JAis restricted to a definite rza~o. This is reasonableas long as evdutio” ;xesses are not taking place. The inapplicability of A3 also has so”~ implicetion for the rale of operation of hierarchical systems as well es their rate of evolution. These will be discussed in the next two sections.

7. SbnIdation

of hiemrcbiil systems

Biological self-reproduction o: more general pro. MSFIPof development may certainly be simulated wib a canventi,mal computer. However,there are es. sential differences hetwen the computer model and ha sy>:em wkachis being modeled. This is true eve” f the interactions in ihe model accurately represent rhe interactions in the real system. Qor example, if we cxnpote the b+navior of a many-body system on the basis of some kirztic model, we do not essume that the ysmm prmexas :afommtion like a 2omp”t.x However. ihe distinction between the hutactions w&h are being simulated and the computation pre

tic computer is more subtle when the system is itself a bona fide information processing system

cess in

operatine: in the hierarchical

mode.

The distinction here arises from the different character of the “description” in classicaland hierarchical systems. We have seen that the classicalself-reproducing system has two types of input: description and external input (data or, more broadly, environment). This corresnonds to ,the DNA and external environ. ment for the cell. In the clssstcal system each manipulable input from the description most be accessedindependently. i.e., the accessing of each such unit m”s* be temporally distinct. This is not ti”e for the biological celt. Certainly there is control over which segments of DNA are accessed at e give” time, but each manipulable unit is not accessedindependeolly, et least 8s far es function (as opposed to translation) is concerned. In fact there are hierarchies of manipulah!e “nits (e.g., nucleotides, genes) which c”n be accessedsimultaneously. The difference between the biolopical ce!: end the

Such :&tg processing systems ect‘in a single le&l mode. Therefore each symbol (representing a nuclectide, amino acid, etc.) mutt be accessedindependently. For example, the interaction between two strings must be determined on the basis of a symbol by symbol comparison. In general this requires a large nombcr of operations. In the cell, of course. this interaction is 8” energy process at the level of tertiary stmcture. Furthermore, this energy process can span many individual enzyme-substrate interactions. This is not possible in the striog processing simulation since only one symbol can be accesseda( a time. Naturally the functions in a present day comp”ter may be altogether more complex the” the simple string processing operations described above. The sys*emmay provide a multiplicity of accessroutes or may allow for accessingmore complicated or distributed inputs-indeed such loal paraUelisms play an important role in determining the rate at which the system can process information. However, all these activities, in particular the examination of the inputs, are subject to the sequential control of the program. If thin were not the case ihe program would not pre. suibe the behavior of the system-essentielly it would

not bc an explicit representation of its transiticln function. This prescriptive coaltrot is absent in bierarchical systems such as the biological cell. The prescribabiiity prapcr?; IS an extra feature, outside the abstract theory of w~tomata. We must pay for this feature, either in the r.;oe domain or by in. creased network complexity Insofar as the latter can be exchanged for time, i.e., innfar a~ tbe functions of the system are not inherently scqucnhal). As a consequence lbe hierarchical system can operate at 3 faster rate (or appear less comtplicatedf than single level systems which perform comparable functiars. This is true even if we u8e the cmgle level system :o represent biologically raalishc mteractions. The% interactions must still be controlled by a subsidiary computation process in the sin& level system. In general systems which act on a prescrinti:; basis (such as tessellation automat?) nr; characterized by this sequential actiai& of the manipulable umts which d*‘::.,~ine their behavior. S?scems which act on XI energy basis (such as the cell1 ian allow for selective control ova their behavior wthaut this sequential feature, but at the loss of effectiveness. This is because the control is acbiemd by t&xin~, an interaction, not by Prescribing behavior in the ordinary sense of pro~amming. Such systems can only be “programmed” by evolution-~:ssentiaUy by trial and error.

I

8.Adiabatic

logics

The disanalogy between lhe DNA description computer programs has importeot implications the rate of evolution. In particular the failure of associated with the fact that the objects which trol the behavior of the cell are determined bv a tion of a manipulable sequerce rather than by quentially accessed input-by a quantity of the

and for A3 is confuncB s:. form

the form qi(r,fo+h,q’... adnuts of gradual modification whereas the latter does not. The sigoificance of this fact is clear in computer programs which embody the mechanisms neceswy for evolution by variation and natural selection. Such progmms have been used for search and optimizstion

H(qr...qlf;

not

of

1966). as well as for studies of the evotwionary behaviw of artificial ecosystems (Conrad and Pattee, 1970). Programs of this type have been useful; for example. in studying various aspects of ecological succession, in part:cular the fixation of genetic traits in an ecologicsl context. In general, however. evolutionary programs do not exhibit learning behavior cornpaiabie to that of natural syrtems-essentially such programs have not sucrceded in combining significant funclion with signiticant rate of evolution. This is because stighr changei in the insirucrions(as opposed IO the parameters) of a computer progx:pm ordinarily produce mayor alterations in the prescribed behavior. lo general a large number of changes are required to produce another functional program. This leads to what Bremermann (1967) has called staantian. or what Minsky (1961) has called mesa phenomena. This delicacy is a feature of the single level mode of information processing, not necessarily of the hierarchlcal mode. In the hierarchical mode the information store constrains certain interactions, often in such P way that a single modification produces only a slight perturbation in the behavior of the system. For example. m an ordinary (single Icvel) switching circuit all the switches are identical. Such circuits are moditied at the level of the connections between switches, not at the level of the switches themselves. This is not true for the cell, where the network of biochemical reactions is determined by enzymes. These are mow complicated ihan on-off elements since they rccognize substra:er as well as switch their transformations. In this case the stl~ctwe of tbc network is determined by the specificity of the enzymes-in contrast to the structure of ordinary switching circuits which in no way depends on the character of the switches. In particular the enzymes may be multifunctional in the sense that they have a catalytic effect on more than one reaction. .4 slight change in enzyme specificity results in a dight change in the extent to which diiferent reactions are catalyzed. In most cases this is

qf(t.t,,+X,?). Theformer

COn3islent with Only a slight change in the Raction network ald ‘herefore with a slight change in the behavior of the system (Godwin, 1969; Kauffman, 1969). There is good evidence that slight changes in 6pcci. ticity often result funn one CT a few genetic chnges.

processes. including applications to game playing and artSlcial intelligence problems (9remerman.n et al.,

In particular a change in one or few amino acids often (but not always) changes the shape ad tb@nfcre

specificity of a” enzyme only slightly (cf. Juncgk, 1971). I” many cases enzymes with different Drimary sequences (&enzymes) perform approximaiely the sane function and in certain cases enzymes can retain considerable furction despite the removal of a num. ber of (noncrucial) amino acids. This fact of gxadudly variable multifunctionality is bnportant since it dlows for the mediation of evolutionary pro:esass by transfer of function. For example, the prhlitivc form of some enzyme may crta lyzs _I ctio,~ A; with a slight change in primary stmctore it may cstzlyze B as well as A; finally, throu& a series o: such G&t changes catalysis of B may become the predomi Iant function of the enzyme. The sieniticmce of thic “adiabatic” variation of function e apropos the rate of evolution, follows from fairly stmple probabilistic >J”stderations (see Appendix). Essentially the maximum possible rate of evolution in. creases with an increase in the number of possible evolutionary pathways each step of which is, to a greater or lesser degee, functiclal or bifunctional. It is important that the degree to which the 34 structwr and therefore function of thy .,.“, ^“*‘lme . ..- ir capable of gradual modification depends on the redundancy of weak interactions which determine this lap. This is itself a” evolutionary property. Nearly all genetic chunges must “pass” through the proteins and ,r ,s thexfore reasonable that the systems which have predominated in the course of evoI&o” ore just those with the apyropriate :edundancy. In this sense the topography of what has bee” celled the adsptivs landscape is itself a” adaptive property. Indeed thts may be the reason for som,e of the apparent excess structore which chamcterizer the enzymes.

9. Co”elmio”a I want to em&size that the hierarchtxl mode of infornmtion process@ is redly wique. It is clear that a hierxm$ical system is not a dig4 system. Neither is it an rnal~ system. In an analog computer the JYP tern is Cndonomicelly) constrained so that its behnvior ii gov.emed by i physical law which is anal* sous to that cf some other system. I” B hierarchical coaputa interactions vary and are seieeted until same appropriate behavior is produced-the modifica. tionr themselves are not induced to achieve the de.

sired behavior. Such induction is infeasible since it would itself require much more information processing than could ever be xpccted fro”1 the system. The paradigm process in hierarchical information processing is molecular folding. We might picture the biological cell as a” information processing system whose behavior is determined t,y the way the pm gram “folds”. There is no anakegous process in any co”“e”tio”el computer. It is clear that both the discrete and the co”ti”uous do play a role in the hierarchical mode. However, a hierarchical system is not a hybrid computer-this is just a system with distinct analog and digital parts which communicate with each other. Essentially the discrete and contir.uous are mixed in the hierarchical mode because msnipulations at the finer, discrete level modify interactions at the higher, continuous level, thereby determininl some reaction network or the way in which the system Loses information about the part. Such infonnatio” processing is not algorithmic or effective in the sense in which we have described ‘MS term. This does not mea” that such sys:cins iso do anything which cannot be done by a single level computation process-the absence of the effectiveness property may have little formal sipiti. cance b this sense. However, it is sigoi&ant in terms of the physical realizability of certain types of behavior. This is particularly true apropos the practicability of evolution-essentldly the difficulty of;lr* gramming a biochemical system to execute some function is associated with the weater oracticebtltty of evolving such a system to perform this funcrion. What is lost in terms of effectiveness of computation is gained in terms of effectiveness of evolution (Conrad, 1969). I think that the fact of such nonalgorithmic computation is redly manifest in our present day concepts of the biological cell-the possibility of such computation has been ignored because of the dtfficulty of handling hierarchical systems es well es the superficial resemblance between coding proewes I” biologtcal cells and conventional computers. I should emphasize, however. that the fact of hierarchical computation does not preclude conventional analog and dlpitnl Information “mcessin~ in biolo&l sw tents. conventionai modes efftcient as Gng as evolution processes are not i”lportant. It Is, of course, possible that various physiolo&al learning

-

l%c

.

ai=

processes-for example, immunologrcal o* neural processes-are based on evolutia. indeed, if such procews are mediated by transfer of functwn they might be feasible only in the hierarchical mode.

Appendix:

slight selective advantage) one-half the WE of the strain from which it derive% The expec:ed number of generations is now

Rates of evolution.

In this appendix I want to illustr:~e the relation between rate of evolution and the genetic distance between different functional states GI’ a system. The fitness of a biological system can be rs~rerented as ui = u,{6il(Pj,G,),Xil) where ui is the fitness of system i. h; is the transition function of system i with states CJ, = (Pi, 6,) and environmental inputs X,. The Pi are rhe phenoiypic states and the Gi = b, b, are the sequence of maw. pulable constituents (nuclrotides) which determine the genotype.The fitness function is a fwxtion of the behavior of the system iu a given environment (Canrad, 1971). ui can change slightly with one or a few mutations if these result in only a slight change in tihe behavior of the system. This is possible becooie chqges in Ci are not comparable to changes in a computer pro gram. Consider the transition Gi ~3 CFl. wth genetic distance m and with uk%= ui. The expected number of gaerations for the appearance of a single system k is Tik =

1 ivopm(l-py-m

where p is the mutation probability, n LSthe length of gene Gi and No is the size of the iuitial population. Here we asmne that the bi can assume just two if they can assume h statesthe expression must be multiplied by (h-1)“‘. There are m! ways in which tltis distance ,:an be traversed in m single step mutations. Suppoce that there is one such pathway for which each U, is corn. parable to or larger than its predecessor. Vie wil! make the simplifying (and unfavorable) assumption that there are no mutations to the next target state until the population reaches (through drift, isolation, or

states;

where W, represents the dead time between the appcarance of a strain and its growth to a point where mutations are allowed. This is the amount of time ln,e are allowing for amplification of the original mutr tions. The ratio -I

_=-

(I

I

+1+N,p(l-,?)“-‘W,)

gives the relative rates of evolution for simultaneous and siepwise jumps of m steps. As an illustration, take N,_= 106, n= 103, and W,= 103. FOIQ= 1O-2, T+jTiiis of the order lo3 ani lo7 for PI equal to 4 and 6. resoectivelv. When D = IO-5 the ra?io increases to 10” and lOit for &se values of m. Naturally these are lower limits since there really is no dead time (but ii is not possible to predict how fast a mutated strain grows without making very particular IIS. sumplions). Also, there may be many possible pathways connecting Gi and Gk (including pathways of more than m steps), multiple mutations are possible even if single step transitions are the most probable, and, of course, there may be many genotypes a;th fitness functioa comparable to that of C;. An analysis on the basis of more realistic assump&ns as well as the relation to the facts of genetics will not be pursued here.

Acknowledgement 1 would lik2 to thank Pmfwsor Behram Kursun~ &I for hospitality at the Center for Theoretical Studies. This research was sup.ported by National Institutes of Health Fellouship Number 1 FO2 CM43960-O: from the Institute of General Medical Sciences and by National Aeronautics and Space Administration Grant Number NGL 10-007-010.

393-409. Camad, M., 1971, Can there be a tbcan of fitness?, to wpcar in: physical Rineiples of Neuronal and “,gnnirmic Pskwior. eL. 1. Conrad and M. b&u (Cmrdon and Breach, NEWYork). Gwnwin. B.C., ,969. A latistical rneeblnics Of temporal owmizatian in ceUs. h: Towwls 3 Theoretical Biology. Vol. 2. ed C.H. Waddingto” U.bdverdty of Ediibuqb Prea,‘Zdi”b”r&) PP. 140-165. hmcgk, J.. ,971, RbDuwinian and non-Darwinian evo,ution ofprorins, Gun. Mod. &al. 3.309-318. Kauffman, S.A.. 1969, Metabolic stnbUity and epigeneds m randomly constructed nets, 1. Theoret. Rio,., 22, 437.-467.