Molecular and evolutionary computation: the tug of war between context freedom and context sensitivity

BioSystems 52 (1999) 99–110

Molecular and evolutionary computation: the tug of warbetween context freedom and context sensitivity

Michael Conrad *Department of Computer Science, Wayne State Uni6ersity, Detroit, MI 48202, USA

Abstract

Proteins and nucleic acids constitute a vast potential reservoir of pattern recognizers that operate on the basis ofshape complementarity. It is possible to construct models of computing in which these shape-based interactionscontribute directly to recognition of signal patterns at the device (or cell) level. The input–output transform is moldedby variation-selection evolution. Such models provide clues as to the organizational features that enable biomolecularmatter to acquire nonevolutionary modes of problem solving through the evolutionary process. The requisiteorganizations are characterized by a high dimensionality that allows them to simultaneously exhibit aspects ofcontext-sensitivity and context-independence. © 1999 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Molecular computing; Context sensitivity; Evolutionary computing

www.elsevier.com/locate/biosystems

1. Introduction

A novel feature of DNA computing is thatindividual molecular characteristics are exploitedfor information processing. This is in sharp con-trast to conventional artificial computing systems,in particular electronic systems, where the keyevents depend on statistical aggregates of atomsand molecules. It is a main characteristic of natu-ral biological systems, however. The major pro-cesses of biochemistry are controlled by theindividuality of proteins and nucleic acids. Thegap between natural and artificial systems, eventhose that exist only on paper or in simulation, isclearly enormous. Nevertheless, the fact that

primitive artificial systems operating on a molecu-lar basis are now being proposed and imple-mented with bona fide molecular materials hasenormous paradigmatic significance, both forcomputer science and for biology.

My purpose in this paper is to focus on fourkey principles: pattern recognition as a basis forcomputing, shape complementarity as a basis formolecular level pattern recognition, macro-microcommunication links as a basis for convertingmolecular recognition to grouping of higher levelsignal patterns, and variation-selection evolutionas a means of molding molecular recognition forhigher level computational functionality. The ob-jective is to use these principles to elucidate therelationship between evolutionary problem solv-ing and modes of problem solving that developthrough evolution. The main claim is this: evolu-

* Tel.: +1-313-5772477; fax: +1-313-5776868.E-mail address: [email protected] (M. Conrad)

0303-2647/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved.

PII: S 0303 -2647 (99 )00037 -4

M. Conrad / BioSystems 52 (1999) 99–110100

tionary search is most effective at creating effi-cient computational functionality if the structureson which it acts are related to the programsembodied in these structures in a nontransparentway. Such evolution-friendly structures may becharacterized in terms of a grammar that exhibitsdual aspects of approximate context-freedom andrequisite context-sensitivity and that uses highdimensionality to compromise these conflictingaspects.

A number of authors have been concerned withthe contribution of context-dependent operationsto the computational capabilities of molecularsystems (Head, 1997; Kari et al., 1997). Our con-cern here is primarily with the role of context inthe structure-function relations that support evo-lutionary adaptation, but the same considerationsbear on computational capabilities. The approachis to develop a model of molecular computing, tobe called the self-assembly model, that incorpo-rates the operative principles noted above andthen to translate the features elicited from themodel into the terminology of formal languagetheory.

2. Pattern recognition: simple versus complex

Each step of a computation can be viewed as amapping from a system’s present state and inputto its next state. In conventional computing ma-chines these mappings are implemented with ele-mental pattern recognizers, such as NAND gates.The pattern recognition action is so simple thatpractically speaking it is generally ignored. Onethinks of the digital computer as being built fromswitches rather than elemental recognizers.

Formal models of computing (Turing machines,partial recursive functions, random access ma-chines, and so forth) all share a common attitudeappropriate to the above picture. The number ofelementary operations should be few and each ofthese should be elemental. Taking complex opera-tions as elemental begs the question.

Biology works in a diametrical manner. Thebase switching components are proteins and nu-cleic acids. Recognition capacity is the main fea-ture. Shape and shape complementarity are the

key mechanisms. Biology, unlike today’s predomi-nant technologies, works with a vast repertoire ofbase recognizers. The prime method of impartingfunctionality to the system is through variation-selection search, roughly by what might be calledevolutionary means of programming, rather thanby the prescriptive programming characteristic oftoday’s machines.

The DNA-based computing scheme imple-mented by Adleman (1994) is the first concreterealization of an artificial system that builds com-putations from complex operations. The mainfeature, for the present purposes, is that the ele-mental pattern recognition operations utilize theremarkable complementary pair formation prop-erties of DNA. The number of possible elementalrecognizers grows combinatorially, i.e. as 4n,where n is the practical limit on the number ofbases.

The possibility of complex pattern recognitionis critically dependent on scale. The recognizersmust be big enough to have complementary fea-tures; atoms are too small. But they must be smallenough to explore each other through heatmotion.

3. Context-dependent pattern recognition

The traditional models of computing mentionedabove have another common feature: they are allcontext independent or at most have very limitedcontext dependency. The logic gates in conven-tional machines implement formally defined in-put–output behaviors that are independent of theoverall design of the circuit (apart from faults).Turing machines read and write symbols on indi-vidual tape squares in a manner that is indepen-dent of the arrangement of symbols on the tape,and so on. Such formal models and the physicalrealizations of them can be quite adequately char-acterized as linear string processing. The newfeature of DNA computing, noted in the previoussection, is that it is possible to recognize andoperate on fairly large strings in a direct manner.The processing is nevertheless still essentially con-text independent since the pattern matching con-sequence of altering an individual nucleotide in

M. Conrad / BioSystems 52 (1999) 99–110 101

the string can be specified without any referenceto the global arrangement of nucleotides.

Biomolecular pattern recognition does not forthe most part have this context-independent char-acter. The recognition capacities of proteins andRNA (at least in some of its phases) depend,roughly speaking, on three dimensional shapecomplementarity. The analogy to lock-key fittingis sometimes used. Alternatively one might thinkin terms of interacting jigsaw puzzle pieces. Thenotable feature is that the relation between thesequence of monomers and function is no longertransparent (i.e. it is not programmable). Thesequence folds to form the three dimensionalshape on the basis of multiple interactions amongmany monomers, including monomers that mightbe quite distant so far as their linear position isconcerned. The contribution of individualmonomers to the pattern recognition capacity ofthe whole depends on the whole. Context depen-dence is the main feature.

The context sensitivity extends to the environ-ment. How the protein or nucleic acid folds de-pends on the milieu. This is inevitable since milieumolecules, whether substrates, control molecules,ions, random metabolites, or other macro-molecules, necessarily interact with the covalentlybonded monomers that officially constituteproteins and nucleic acids. The recognition-actioncapabilities of a macromolecule cannot at thesame time be made sensitive to particular milieufeatures and isolated from others without consid-ering the cost of building this selectivity into it.

The lock-key metaphor is in reality too simple.The docking of an enzyme with a milieu moleculeor with another macromolecule is a highly dy-namic process. Conformational dynamics is criti-cal. The structure-function relation is morecomplicated than solving the extremely difficultproblem of calculating the 3-D shape of themacromolecule from its linear sequence ofmonomers. The recognition capacity of the ele-mental biological operators is even greater thanwould be anticipated on the basis of a purelyshape fitting model. The metaphor of dynamicshape fitting is more realistic.

4. Conformation-driven computing

The recognition-action problems presented toorganisms and machines ultimately involve inter-actions with the macro world external to them.Appropriate output actions must be taken in re-sponse to impinging patterns of input signals. Ifindividual molecular characteristics are to con-tribute to this input–output transform, then it isnecessary to transduce the external input into aform that the molecules can sense. At some stageit is necessary to amplify the molecular level re-sponse to yield a macroscopic output. This is thegeneral situation with molecular devices (such asbiosensors). If all the molecular level recognitionpower could be summarized in terms of a fewparameters (e.g. reaction on or off) it would bepossible to ignore it and nevertheless account forthe signal processing capabilities at the macrolevel. Can the transduction be done in a way thatmore fully exploits the pattern recognition powernative to proteins and nucleic acids—that allowsit to show through at the macro level?

The thought device depicted in Fig. 1, to bereferred to as the self-assembly model, is intendedto show that molecular level pattern recognitioncan in fact percolate directly to higher levels(Conrad, 1990a, 1992a). The signal lines imping-ing on the device are either in an active (1) or

Fig. 1. Schematic illustration of self-assembly model of com-puting.


inactive (0) state. Each line is associated with adifferent protein, which is released if the line isactivated. Input signals which have no shapeproperties are thus re-represented as molecularshapes. These molecular shapes self-assemble, likeself-organizing jigsaw puzzle pieces, to form acomplex. Different complexes, each with a varietyof shape features, form in response to differentinput patterns. Enzymes that recognize shape fea-tures associated with a particular input patterncan then trigger an output action appropriate tothat pattern. More significantly, shape featurescommon to different complexes can be associatedwith different groupings of the input patterns.

The impinging symbolic pattern recognitionproblem is thus converted to a free energy mini-mization process. All the interactions that con-tribute to macromolecular self-assembly arebrought to bear, including interactions with com-ponents of the milieu. The molecular level patternrecognition capability is not compressible into aspecification that is independent of the underlyingphysics in a manner comparable to the context-in-dependent switching operations realized by thebase components of a digital computer. Thedevice level pattern recognition capabilities—theinput–output transform—must take account ofthe underlying physics in a way that the engineerrenders unnecessary in the digital case. Shape-based molecular pattern recognition percolates upto higher level symbolic pattern recognition.

The type of processing illustrated by the self-assembly model does not require the self-assemblyof distinct macromolecules. The conformationaldynamics of a single protein (or of an alreadyself-assembled complex) would be sufficient tosupport higher level pattern processing. Picture,for example, a device in which the input signalsare transduced into milieu features (e.g. they trig-ger the release of a particular ion, a particularcontrol molecule, a change in temperature orsome other general physiochemical condition).Different signal patterns are thus transduced todifferent milieu patterns. The pattern sensing en-zymes respond differently to these different pat-terns (or milieu contexts). Interactions withparticles of the milieu modulate the folded shapesof these enzymes (since they mix in with the

interactions among the electrons and atomic nu-clei that define the molecule). Different input sig-nal patterns, by virtue of being transduced todifferent milieu patterns, will therefore lead todifferent shape features. Readout enzymes thatrecognize a shape feature associated with a partic-ular grouping of the input patterns can then trig-ger reactions that lead to an appropriate deviceoutput. Again shape-based molecular patternrecognition percolates up to the higher (device orcell) level of symbolic signal processing.

In natural biomolecular systems every enzy-matic reaction is accompanied by conformationchanges. We can picture enzymes as processingnetworks consisting of thousands of electrons andatomic nuclei that fuse milieu influences in ahighly conformation-dependent manner to yieldshape changes that finally control metabolic trans-formations. The network of metabolic transfor-mations itself serves, through reaction anddiffusion, to fuse signals in space and time. Theself-assembly model is intended to illustrate thatthis metabolic level of processing is not the wholestory: cell-level capabilities cannot be fully ac-counted for in terms of reaction-diffusion modelsthat ignore the powerful context-sensitive fusionof milieu influences afforded by processing at thelevel of individual molecules.

In principle networks of conformation-drivenprocessors could be constructed to have generalpowers of computation. This follows from the factthat networks of formal neurons that only aver-age the patterns of pulses impinging on them canbe constructed to implement any finite automaton(McCulloch and Pitts, 1943). A fortiori it shouldbe possible to do the same with a suitable reper-toire of more powerful pattern processors, and todo so more efficiently. The problem is to generatethe repertoire and to orchestrate it into a coherentgrouping. But the goal need not be to duplicategeneral computational capabilities. It would bequite useful to develop pattern processors specificfor particular data fusion tasks.

The self-assembly processor could also be im-plemented with DNA (Conrad and Zauner, 1997,1998). The basic idea is to code 0’s and 1’s intomethylated and unmethylated DNA strands thatthen bind to a backbone. The pattern of input


signals is converted to a pattern of methylation onthe hybridized duplex. Secondary structural fea-tures of the duplex, in particular the equilibriumof B and Z DNA, would be detected spectroscop-ically (by circular dichroism).

5. Substrates of evolutionary search

The chief feature of the conformation-drivenmode of computing described above is its utiliza-tion of the self-organizing dynamics of individualmacromolecules to integrate milieu signals. Thetransduction processes that convert external signalpatterns into milieu patterns can provide addi-tional reaction-diffusion mechanisms of signal in-tegration. Such systems are in their very naturerefractory to conventional programming. Learn-ing approaches are necessary. The situation issimilar to that with connectionist neural networkmodels. The potential surface is generally moldedby an adaptation procedure, e.g. methods actingon individual networks through error feedback(Werbos, 1993), simulated annealing (Kirkpatricket al., 1983), and population-based evolutionarycomputation methods (Kampfner and Conrad,1983; see Fogel, 1998 for a historical introduc-tion). The difference is that neurons that serve asthe base elements in connectionist networks havea fixed repertoire of capabilities, whereas themacromolecules that serve as the base elements ina conformational processor constitute an evolu-tionarily extensible repertoire.

Three basic types of adaptation procedures arepertinent:

1. Variation-selection evolution of the macro-molecular components of the processor. In theself-assembly model described in the previoussection this would mean directed evolution ofproteins entering the complex or responding tothe milieu signals as well as evolution of thereadout enzymes that recognize the resultingconformation features. In natural biologicalcells the macromolecular pattern recognitionwork could be distributed over multiple recog-nition events associated with multiplereactions.

2. Variation-selection acting on the coding ofinput signals into milieu features. This appliesto the variant of the self-assembly model thatutilizes the conformational dynamics of a sin-gle protein or complex. The procedure wouldbe to choose a naturally occurring macro-molecule or complex and determine how itsactivity depends on different combinations ofmilieu features. Spectroscopic means could beused to identify different input signal patterns.Altering the coding of input signals into milieufeatures could also take place in natural bio-logical cells. The target of evolution wouldthen be the enzymes that control the transduc-tion dynamics.

3. Adaptive self-stabilization. Here error feed-back signals build an adaptive capability intothe conformation-driven processor. When theinput–output transform performed by the pro-cessor fails to satisfy required criteria an errorsignal alters the distribution of proteins in away that persists due to self-assembly proper-ties. Structure modification continues until theerror signal is attenuated. Error feedback act-ing on structure allows for adaptation to occurwithout requiring a population and to do so inan ongoing manner.

6. Importance of high dimensionality

Variation-selection search can be viewed as aform of computation itself (cf. Fogel, 1995). Thesearch algorithms are defined by the variationoperators used (e.g. mutation, cross-over), by theselection regime (e.g. elitist, in proportion tofitness, tournament selection), and by the mannerof specifying fitness criteria. Subsidiary strategies,many motivated by the classical theory of specia-tion, are sometimes employed. The results of thiscomputation could be the solution to a particularproblem (say an optimization problem) or it couldbe a computational system that solves some classof problems. The latter would be the case if theobject of adaptation is a neural net. Clearly it isthe case with a conformational processor.

The organization on which variation and selec-tion act should also be considered, since some


Fig. 2. Mutation-buffering redundancy (left) and corresponding extradimensional bypass (right).

organizations are more and some less moldable.Evolutionary computations implemented on digi-tal computers make different choices in this re-spect, some emphasizing genotypic and somephenotypic representations. If the models weretrue to nature the mapping between genotype andphenotype would be the most important consider-ation. The folding of a protein is the simplestexample. The sequence of amino acids is thegenotypic representation; the folded shape is thephenotypic representation. Variation acts on thegenotype, while selection acts on the phenotype.The importance of the genotype-phenotype map-ping is that it increases the probability that rea-sonably likely genotypic variations will produceacceptable phenotypic changes.

The reason can be schematically illustrated by aballs and springs analogy (Fig. 2). Picture thecovalent bonds that order the sequence of aminoacids in a protein as strong springs; picture theweak bonds (e.g. van der Waal’s forces) that areresponsible for folding as weak springs connectingmultiple different amino acids. Different aminoacids are represented by balls of different sizes.The mutational replacement of one amino acid ata critical site (say near the recognition site, repre-sented in Fig. 2 by the unlinked neighboring pairof amino acids) would likely have an unacceptableeffect. The ramification of most mutations will bedistributed around the network, thereby allowing

for a buffering of the distorting effect on thefolding dynamics. If the buffering is insufficient toallow for acceptable mutations more noncriticalamino acids can be added. Or amino acids withcloser structural analogs can be used. Near thecritical sites the effects of mutation are morecontext sensitive; away, where they are morenearly context free, redundancy plays its bufferingrole. Redundancy overlaid on self-organizing dy-namics yields evolution friendliness.

The situation is pictured geometrically, in termsof fitness surfaces, on the right hand side of Fig.2. In principle this should be a multidimensionalplot of fitness (some measure of performance)against gene structure (in the present against asufficient number of axes to specify the sequenceof amino acids). The necessary condition for evo-lution to proceed (rather than stagnate) is thatdifferent acceptable points on the surface bereachable by single mutations. Stagnation resultsfrom the fact that the time required for movingfrom one acceptable point to another scales in-versely as a product of probabilities if specificsimultaneous mutations are required (for calcula-tions see Maynard Smith, 1970; Conrad, 1972,1983). Multiple simultaneous mutations (or othervariations) of course occur and may be associatedwith increases in fitness. But if high points areseparated by deep valleys in all dimensions insuch a way that two or more specific mutations


must occur, or if one and no other must occur,then these will be points of evolutionary stagna-tion. Such surfaces, or landscapes, are frequentlyreferred to as being rugged.

If the dimensionality of the space is increased(by increasing the number or variety of aminoacids used) the chance that isolated peaks will beconverted to saddle points increases. The peaks inthe lower dimensional space are thereby mappedinto corresponding saddle points in the higherdimensional space, i.e. into points that afford anuphill pathway that can be traversed in singlemutational steps. The evolution time then scalesas a sum of single step evolution times, with thescaling factor largely depending on the fraction ofupward running pathways. Under a wide varietyof conditions this fraction increases with dimen-sionality. The fitness landscape is smoothed.

Pathways of evolution that appear as a result ofincreased redundancy can be thought of as ex-tradimensional bypasses. Formal arguments sup-porting this conclusion may be developed fromMay’s analysis of complexity–stability relationsin dynamical systems (Conrad, 1990b; cf. May,1973) and from information theory analysis of theeffect of primary sequence on tertiary properties(Conrad, 1979, 1983). The conclusion is also sup-ported by empirical analyses of structure–func-tion relations in hemoglobin and immunoglobins(Conrad and Volkenstein, 1981; Lara-Ochoa etal., 1996) and by computational experiments(Chen and Conrad, 1994). But for the presentpurposes the basic intuition is sufficient: if thetopography of the fitness landscape is randomthen it is increasingly likely that isolated peakswill transform to saddle points as the dimension-ality increases.

The advantage of biomolecular materials forevolutionary computations, as compared to con-ventional programmable architectures, is that theynaturally embody the type of structure-functionplasticity that supports evolution friendliness.Computer programs per se are extremely fragile inthis respect. The plasticity conferred by proteinfolding might in principle be represented at thevirtual level (i.e. at the level of simulation), butthe computational costs of doing so would beenormous. The creation of extradimensional by-

passes by the addition of redundant material thatis to some extent superfluous as regards mechanis-tic function is a modest cost. In nature suchredundancies hitchhike along with the advanta-geous traits whose appearance they facilitate(Conrad, 1983).

7. Place in the Chomsky hierarchy

Now we can return to the original claim: evolu-tionary search is most effective at creating effi-cient computational functionality if the structureson which it acts are related to the programsembodied in these structures in a nontransparentway. Such evolution-friendly structures may becharacterized in terms of a grammar that exhibitsdual aspects of approximate context-freedom andrequisite context-sensitivity and that uses highdimensionality to compromise these conflictingaspects. The claim is a characterization, in termsof formal language theory, of the material organi-zations that can support the recognition andadaptive attributes necessary for conformation-driven computing. Basically it is a translation ofterms that have so far been used in a ratherinformal manner: structural nonprogrammability,context dependent pattern recognition, mutationbuffering and high dimensionality. To the extentthat the conformation-driven model abstracts keyfeatures of biological information processing thecharacterization extends to organisms.

To indicate how this translation works let usbriefly recall some basic concepts. The conforma-tion-driven system has a set of states and maps(or transition functions) determining the state tostate transitions given the present state and input.These maps, which can be written out as a stringof symbols, constitute the program of the system.The set of input sequences that trigger a selectedoutput action (or that bring it to some selectedsubset of its states) is referred to as the (formal)language that it accepts. Our immediate concern,however, is the relationship between the programand the structure that implements it. We assumethat the amount of componentry available, whileit may not be fixed, is subject to limitations. Wewill also assume, to avoid issues beyond our


present scope, that the processor can be ade-quately modeled as a discrete state, discrete timesystem.

Let us also recall the concept of a grammar (seee.g. Hopcroft and Ullman, 1979). This is a set ofpermissible substitutions of the symbols on astring that yields a language. The generative pro-cess begins with some initial string that includes atleast one symbol that admits a substitution (i.e. atleast one nonterminal symbol), and then proceedsthrough a series of steps up to termination (if thisever occurs). The rules for permissible substitu-tions are referred to as productions. In the case ofa natural or computer language intermediate sym-bols (such as ‘verb’) are used, while words such as‘run’ would be terminals. For the purpose ofdescribing a system’s structure (but not necessar-ily its behavior) we consider a special case, to becalled a mutational grammar, in which there areno intermediate symbols and no symbols are ulti-mately terminal. Thus the possible evolution of aprotein would be described in terms of possiblereplacements of amino acids and also the interac-tions among them. The grammar is context free ifthe substitutions that are allowable for the sym-bols in a string are independent of any othersymbols in the string. Otherwise we will say thatthe language is obligatory context sensiti6e, mean-ing that it belongs to the complement of thecontext free languages (since we want to excludethe context-free subset and want to include appro-priate languages that belong to bigger languageclasses). The grammar will be called globally con-text-sensiti6e if the number of productions in-creases with the number of symbols in the string.But we will assume that a finite grammar canwork adequately well in this case, either becausethe number of components is limited or becausenew interactions among them that enter at acertain point are weak enough to be ignorable.

Suppose, for example, that we represent aprotein by the string

ai(1)aj(2)…ak(n-1)al(n)cp(1,1)cq(1,2)…cr(1,n)

…cs(n,1)…ct(n,n-1)cu(n,n)

where the symbol ah(m) specifies that position min the sequence of strongly (covalently) bonded

amino acids is occupied by an amino acid of typeh and the cg(m, f ) specifies that amino acid m andamino acid f are connected by a weak (noncova-lent) interaction that can be classified as being oftype g (including null interactions). The ah(m) canbe mutated independently of one another, apartfrom bookkeeping changes in the positionalmarker m that are necessitated by additions ordeletions. But it is not possible for them to mutatewithout alterations in at least some of the cg(m, f )that cannot be automatically accounted for on thebasis of bookkeeping. The grammar is globallycontext sensitive since the number of alterationsin the cg(m, f ) concomitant to a mutation of anah(m) increases with n.

We consider two languages, to be called theterritory language and the map language. A stringof symbols in the territory language is a directcoding of the structure of the system in terms ofcomponents and wires connecting them (in factfor the experimenter it could be the structureitself). As indicated above, the components in thecase of proteins and nucleic acids would be theconstituent monomers (and in reality some milieuatoms or molecules). The wires connecting themwould be the strong and weak interactions amongthe monomers (ultimately photon exchanges). Inthe case of a digital circuit the components wouldbe the switches and the wires would be the elec-tron conduits connecting them. The map is aprogram used to generate the state to state behav-ior of the system; it is a function in the mathemat-ical sense. If the system is described by anequation of motion the map could be a programused to solve this equation. The grammar thatgenerates the map language need not be a muta-tional grammar.

The territory grammar is context-free if allcomponents and wires can be added, deleted, orreplaced according to some set of rules (the syn-tax), with the provision that all the rules arecontext-free. The map grammar is also context-free as long as all the rules for generating anymap are context-free (but we do not require thatdifferent maps can be directly transformed intoeach other in a context-free manner). If the terri-tory and map grammars are not context-free wewill take them as obligatory context-sensitive. If


the number of components is finite and if the stateset can be approximated by a finite state set, thenboth the territory and map languages could inprinciple be generated by an even more restrictedgrammar (the regular grammars), since theywould be the languages associated with finite au-tomata. But the description would be much morecomplicated in the absence of any natural rela-tionship between the rules and the mechanisms.

We can now define structural programmability(or transparency) more carefully. The relation be-tween a system’s structure and the function thatdescribes its state to state beha6ior will be consid-ered programmable if the territory and map gram-mars associated with it are context-free and ifsuperprograms (cross-compilers) could in principlebe constructed to translate the territory and maplanguages to each other in a finite number of steps.If these conditions fail the structure–function re-lation is non-programmable. They would fail inthe protein case since it is quite infeasible tomutate a single amino acid without changing thewiring (interaction) structure of the protein at thesame time. When both components and wires aretaken into account the transformation of stringsto strings is highly globally dependent. Some ofthe modified interactions could be quite impor-tant, others insignificant enough to be ignored.This is where approximate context freedom enters(but we will not here attempt to put a measure onthis).

If the territory grammar is context sensitive themap grammar will be as well. This expresses thefact that there are more equations that satisfy thesyntactic requirements of mathematics and thegeneral principles of physics than equations (in-cluding boundary and initial conditions) that arefeasible. To determine what is feasible it is neces-sary to perform an experiment or to solve theequation itself (i.e. execute the program) to see ifit is tenable and agrees with the phenomenology.This in general is not a finitistic process, andtherefore does not satisfy the translation condi-tion for structural programmability. The extremecase is that of structural programmability itself.Every physical system is globally context-sensitivefrom the point of view of the underlying physics(since in principle all particles interact with each

other). The engineer must work hard to suppressthe global context dependencies to create a secondlevel territory that is sufficiently context-free toapproximate structural programmability. To theextent that this is achieved the distinction betweenthe territory language and the map language be-comes one of convenience only. The physicalequations can be entirely dispensed with. Butclearly it would be impossible to use the equationsof physics to establish in a finitistic way that thesesame equations are irrelevant. The relationshipbetween the base territory language (i.e. the un-derlying physics) and the relevant map language isso nontransparent in the case of a structurallyprogrammable system that the base language canbe dispensed with altogether.

The conformation-driven processor, and by im-plication biomolecular systems generally, sits inan intermediate zone. The significance for evolu-tion can be found here. In general no metric canbe put on the degree of change exhibited by thesequence of states generated by a mutated pro-gram. At very least such a metric, if it existed,should make it possible to determine whether thesystem runs to an absorbing state or continues tochange. The unsolvable halting problem would besolved. But this is in general. Structurally nonpro-grammable systems, because of their obligatorycontext sensitivity, allow for more peculiarlystructured map languages than programmablesystems. The context sensitive rules can work tocreate subdomains with either unusual gradualismor erraticism. What appears as the mutation of asingle component (amino acid) in the territory lan-guage is really a compound substitution for thatcomponent together with wires (interactions amongamino acids) that depends on multiple other compo-nents and wires. In short the system folds. Thefolding, or more generally self-organization, canbundle structures that are separated by single mu-tations into maps that yield closely related state-to-state beha6iors. The hidden interactional changesthat are concomitant to the 6isible component mu-tations do the job. But to obtain the topologicaldistortability necessary for gradual (as opposed toerratic) function change, buffering is necessary.This means the mutations must occur in a nearcontext-free way. The context sensitivity yields the


self-organization; the context freedom allows forthe transformability of that organization. Again,following the mutation-buffering model, high di-mensionality is the compromising factor. As thedimensionality increases the contexts relevant tothe mutation of any given type of amino acid canbecome more differentiated.

Finally we can consider efficiency from thepoint of view of language acceptance. This iscloser to function in the normal biological sense,i.e. what the system does to the input signalpatterns impinging on it. We can again look atthe self-assembly model. The concern so far hasbeen with translating the folding and redundancyconditions for evolution-friendliness into the ter-minology of context sensitivity and context free-dom. The input-output capabilities of theself-assembly processor depend on the same fac-tors. Signals are fused by conformational self-or-ganization (an extension of the dynamicsresponsible for folding). High dimensionality al-lows for a combination of high sensitivity toselected milieu contexts and low sensitivity toothers. The synergy of mutation-sensitive and mu-tation-buffering domains becomes a synergy ofdomains sensitive to critical milieu features anddomains that buffer the effects of noncritical fea-tures. The same tug of war between context sensi-tivity and context freedom that yields highevolutionary adaptability also confers high com-putational efficiency.

To get a sense of the potential power let n equalthe number of components and C be the numberof wires with which each component can maxi-mally be associated. If the system is structurallyprogrammable then the addition of new compo-nents cannot be allowed to introduce new interac-tions that would alter the contexts under whichthe input–output behavior of the existing compo-nents is defined. The number of interactions avail-able for problem solving is thus Cn. If the systemis nonprogrammable then the addition of newcomponents will in general increase the number ofinteractions available. If each component were anindividual particle the number of possible interac-tions could grow maximally as fn2, where f is thefraction of operatively significant interactions.Context sensitivity increases f; context freedom

decreases it. The number of interactions availablefor problem solving in the structurally non-progammable system grows fn/C times faster thanin the structurally programmable system. Even ifn is limited and f overall is small and decreasingwith increase in n, this would be fast enough tohave a substantial real-time impact on the size ofpolynomial-time problems that could be ad-dressed. This is in an entirely classical picture.Quantum features could conceivably speed upmolecular recognition, due to the influence ofelectrons on conformational dynamics (Conrad,1992b). If so pattern processing capabilities of aconformation-driven processor would require ex-ponential digital resources for their effective simu-lation (cf. Lloyd, 1996).

8. Concluding remarks

We can summarize thus: context-dependencecan be correlated with self-organization; self-orga-nization can be interpreted in terms of context-de-pendence. The monomers in a macromolecule andthe photon exchanges between them (the strongand weak bonds) cannot be altered independently.Let the monomers and links among them berepresented as a string. Rarely would it be possi-ble to change one of the monomeric elements inthe string without changing a number of the linkelements. The changes that are tied together serveto bundle the system into dynamical regimes withatypical properties, in particular either gradual orpunctuated dependence of behavior on structurechange. The degree of context-dependence de-pends on how many elements necessarily changetogether. If the context-sensitivity is extremelyhigh this is good for highly specific interactionswith other molecules; but it is also likely to meanthat so many elements must change together thatthe sensitivity to mutation will be too high forevolutionary adaptation. If the context sensitivityfalls to nil the self-organizing tendency is lostaltogether since this means stripping away theinteractions intrinsic to the monomers. Low (notnil) context regions buffer the effect of mutation.When a monomer in one of these regions changes,the interactions that change with it just slightly


add to or decrement the interactions bearing onthe high sensitivity region. For such bufferingeffects to occur many monomers that approachcontext independence must be included in thesystem, hence the importance of highdimensionality.

Even DNA, though specialized as the genotypicmolecule, fits to this picture. Complementarypairing occurs in a rather context-independentway. But DNA is not a perfect helix. Secondarystructural effects, including reversals of handed-ness and conformational polymorphism bothwithin right and left handed forms, are dependenton both sequence and milieu conditions (Sa-sisekharan, 1983). Such conformational featureshelp with readout and contribute to gene regula-tion. Redundant DNA, therefore high dimension-ality, is a prominent feature of the metazoangenome. Regulatory change is probably at least asimportant for metazoan evolution as the evolu-tion of new protein functionalities. The redundantDNA serves naturally to buffer the mutual effectsof sequence variations in different coding andregulatory regions (Conrad et al., 1985). Context-independence (the complementary pairing fea-ture), context-sensitivity (the dynamicalself-organization of shape features pertinent toregulation), and high dimensionality (the presenceof introns) work together to yield structure–func-tion relations that are suitable for evolving aprocedural mode of processing involving the se-quential inhibitory and activating action of geneson other genes. Different models of artificialDNA computing have abstracted different DNAproperties: pattern matching, splicing, secondarystructure dynamics. In the natural biological sys-tem these different modes work in concert to yieldpowerful synergies between evolutionary, dynami-cal, and procedural modes of problem solving.

Acknowledgements

This research was supported by the NationalScience Foundation under grant ECS-9704190. Iam indebted to K.-P. Zauner for valuablediscussion.

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Molecular and evolutionary computation: the tug of war between context freedom and context sensitivity