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Medical Hypotheses 17: Zlq-Zag, 1985 A THRESHOLD MODEL FOR PUNCTUATED GRADIIALISM J.A. Kieser. Department of Orthodontics, School of Dcnti xtrq . Xi twatcrsrand University, Johannesburg, South Africa. H.T. Groencveld, Department of Biostatistics, Medica 1 Rcscarch i‘ouni, i I Johannesburg, South Africa. ABSTRACT A model for continuous and discontinuous evolutionary chanec is pt~opo~~~d that accommodates both punctuated equilibrium and phyletic gradualism. Natural selection operating on a gaussian distribution of phcnotypc5 subjected to severe stress will result in a shewine of the normal dist-r-ihutIf a reproductive isiolation ion in the direction of the favoured phenotype. threshold is imposed upon this model. r a p i d crossing; of the t h r e s h o l d by large numbers of individuals within the skewed distribution wi 11 r<~~ul t in the sudden achievement of a critical descendant population ma\\. Yild directional se1 ection however, w i l l result in gcntlc s h i f t s of’ the qaussian mean thus leading us to suggest that stasis and incipient .pcciatio are extremes of the same spectrum, defined by the rates of mran-shift in response to varying severities of selection pressure. Both pun\,tuated and gradualistic evolution as accommoda.ted in our model may furthct- bc described within Thorn’s cusp catastrophe theor->-. Though the process of evolution is frequently employed b>- biomedical scientists to address questions of phenotypic change and adaptation in populations rangeing from micro-organisms to man, few have questioned the underlying conceptual mechanisms of evolution. S i n c e ilhar-lcs D,~rwin first stressed the importance of ‘I... slow and gradual accumulation of numerous slight, yet profitable, variations” (1). the model of phylctik. gradualism has survived the neo-Darwinian synthesis to gain asccndanc> as the dominant model of evolutionary change (2. 3, 4. j, 6). Significant questions concerning this mechanism have recently been raised with the formulation of the model of punctuated equilibrium by Eldredge and Gould ( 7 , 3?, which has in turn prompted the “decoupling” of macro-cvslut i,>n from micro-evolution (9, 10). In this communication, we suggest a unitary model of evolution which accommodates both gradual unidirectional phenotypic change and sudden speciation followed by stasis. THE THRESHOLD MODEL The normal distribution density of phenotypes in a randomly mating population may be characterised by (or transformable to) a gaussian distrihution with average phenotypes clustered about the mean and phenotypic divcre;ents 219 peripherally located in the tails. Anagenesis within this population is governed by natural selection which tends to maintain symmetry around the mean under stable environmental conditions. Gradual phenotypic change due to progressive adaptation or to adjustments to gradual environmental change are achieved simply by a shift in the gaussian mean. In order to explain speciation however, it is necessary to impose a threshold on the normal distribution such that only individuals who lie a given distance from the mean (measured in standard deviation units) may achieve reproductive isolation by crossing over the threshold (Fig. 1A). Phenotypic peripherals that cross this reproductive isolation threshold under stable environmental conditions perish because they fail to achieve the critical population mass necessary for the establishment of a viable descendant population (normalising selection). Crossed over individuals might be considered as equivalent to Goldschmidt's hopeful monsters (ll), or Schindewolf's grossmutants (12). Under environmentally stressful conditions, the rapid accumulation of phenotypes that will allow their possessors to survive in preference to those phenotypes that render them more liable to extinction will skew the normal distribution in the direction of the favoured phenotype (Fig. 1B). ( I A B Figure 1 A threshold model of speciation in a randomly mating population: A - gaussian distribution of phenotypes under stable environmental conditions; B - skewing of the distirbution in response to severe environmental stress. If clustering is directed towards the threshold, a situation will be reached where large numbers of phenotypic favourites will have accumulated just short of the threshold, accompanied by very little movement in the population mean (since the mean continues to be influenced by extreme values). One result of this disproportionate accumulation of individuals is an increase in the relative abundance of mutations short of the threshold, a "genome in turmoil" (13). Further skewing spurred by the continuous erosion in the numbers of unfavoured phenoytpes and possibly precipitated by a rapid accumulation of mutants will result in the rapid crossing over of large numbers of favoured individuals to reproductive isolation, 220 esulting in turn in the achievement of a critical population mas, cf‘tcn shenotypically vastly different from its ancestral population (FiL:. 2). Figure 2 Further skewing of the normal distribution as a result of nitural selection results in rapid crossing of large numbers across the rcproductivc isolation threshold and hence the establishment of a critical population mass. Whilst persistent or lethal stress will render the putative ancestral population extinct, a progressive lessening threat will result in the formation of two chronospecies. In the absence of a threshold. the entire species may either become extinct or undergo gradua 1 phyletic change. depending on the severity of the threat. PUNCTUATION ANP GRAIXIALISM The punctuation model holds that most evolutionary change is concentrated in geologically instantaneous speciation events (7, S;, 14). Al though the model thus defines the mark of origin of a species it offers no acrual mechanism of speciation (15). Vrba has suggested that the spcciation model favoured by punctuationalists is that of allopatric sprciation in small isolates (10, 17). Against this sudden origin of species is set the more conventional theory of phyletic gradualism which strcs\ch the gradual transformation of one species into another (3) and ernphd?i<.eq the lack of spatio-temporal discreteness of species (16. 15). Punctuat ionalists originally based their model on the rapidity with which. mor-phological discontinuity had been shown to appear in the fossil record ,rnd on the observed lack of transitional fossil forms. which has r-eccntlr been underscored by studies on Ordovican trilobites, Mesozoic and Cenozoic molluscs. radiolarians and hominids (19, 20). D e t r a c t o r s however , lia\rc pointed out that the punctuation model is based upon negative rather than empirical evidence (gaps in the fossil record) (Zl), that the model is based on a temporal solipsistic definition of the graduality of “zraduat (22) and that rapid phyletic speciation may be accounted for by Lewis’ saltation model or Whites’ stasipatric speciation model (10). Based on Ozawas ’ observations G!.j)? we argue that time-dependent uniditol~tion;tl 221 phenotypic change (gradualism) coexists with rapid speciation as visualised by the punctuation model. That punctuation and phyletic gradualism are not mutually exclusive has recently been shown in Cretaceous ammonites and Jurassic bivalves where a punctuated pattern of morphological discontin uity coexists with gradual changes in size and sutural complexity (24, 25). CUSP CATASTROPHE THEORY Our punctuated gradualistic model may be better visualised using Rene Thorn's qualitative catastrophe theory. Briefly stated, the model relates to a system governed by a potential function V(QiAJ) where 83 is a set of k externally controlled parameters and Q. a set of n state variables or generalised coordinates. Whilst minimalivalues of Q. govern stability, In a kstationary values of V with respect to Q. govern equilibrium. dimensional space, surfaces may then be defined by the n equilibrium equasions av _ -_=u aQi whilst lack of equilibrium may be expected at critical states defined by the vanishing stability determinant a'v (26, 27). aQiaQj A topological speciation model may then be plotted with the two control parameters reproductive success and restraint as axes on a horizontal plane which is referred to as the control surface (Fig. 3). For each point on this surface the most probable behaviour of the species is represented by a point directly above that on the control surface at a height appropriate to the behaviour (28). Collectively, these behaviour points present another surface above the control surface called the behaviour surface. This surface is characterised by an overall slope from high to low levels of reproductive success (Fig. 3). Rene' Thorn's cusp catastrophe theory indicates that in the middle of this surface there is a smooth double fold defined by the bimodality of behaviour at some of the control points. The fold lines of this pleat define catastrophes or sudden irreversible changes in behaviour. Consider path 1 in Figure 4, a species subjected to mild stress will experience a gradual gaussian mean shift due to directional selection towards higher levels of reproductive success (phyletic gradualism). Path 3 however, indicates the behaviour of a threshold distribution. The stressed population moves on the lower behaviour surface towards point B when it encounters the fold line and jumps the gap towards the upper level of the behaviour surface - completing a speciation catastrophe (punctuated equilibrium). A reversal of this path (path 2) leads to a sudden extinction catastrophe. HIERARCHIALISATION AND SPECIES SELECTION The spatio-temporal individualisation of species offered by our threshold model allows for the consideration of species as units of selection, with favoured units exhibiting high speciation rates (7, 9, 14, 20, 29). Within our model, species selection will favour specific variability and the formation of interspecific thresholds, which in turn will lead 222 to higher speciation rates (considered to be the prime mechanism of specie? selection) (14). Renk Thorn’s generalised cusp catastrophe graph (see text for Figure 3 explanation). Figure 4 Continuous and discontinuous changes in speciation and extinction shown as paths on the catastrophe graph (see text for explanation). P<,th 1: phyletic gradual change, Path 2: extinction, Path ,?: p u n c t u a t e d equilibrium. 223 MUTATIONS According to our model, the sheer bulk of pre-threshold individuals within the skewed gaussian curve will result in relatively high numbers of mutations (gene - or chromosomal) available to the population prior to speciation. Vrba has in fact suggested that the quality of change during the speciation event may differ from that which characterises gradual change (16). We hypothesise that this relative accumulation results in a mutation pressure that enhances threshold crossing and may result in the establishment of a descendant species that will differ quite considerably from the ancestor-al phenotype. We further support the hypothesis that selection will favour species that code their genetic information so that mutations are more likely to result in viable individuals - selection for selectability - thus giving rise to a double feedback loop cybernetic system (30). It is also suggested that the relative increase in mutations in the pre-speciation distribution will result in an increase in phenotypic variation during speciation, an effect that has recently been documented in Cenozoic molluscs (31). EXAPTATIONS Gould has stressed the importance of those features that did not arise as adaptations but are by-products of selection (14, 32). We suggest that because of the high concentration of potential mutations in the skewed pre-speciation gaussian curve, large numbers of exaptations (nonadaptive features) may be "piggy-backed" across the threshold, thus further enhancing the phenotypic differences between descendant and ancestral populations, and possibly providing a way out of Mivart's dilemma. REFERENCES 1. ;: 4. 5. 6. 7. 8. 9. 10. 11. Darwin C. On the Origin of Species by Means of Natural Selection. Murray, London, p. 454, 1859. Berry RJ. Neo-Darwinism. Edward Arnold, London, 1982. Gingerich PD. 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