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The ontogeny of morphological variation : an example from yellow-cedar [Chamaecyparis nootkatensis (D.… Banerjee, Satindranath Mishtu 1990

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THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR [Chamaecyparis nootkatensis (D. Don Spach)]. by Satindranath Mishtu Banerjee B.Sc, University of British Columbia, 1986. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Botany We accept this thesis as conforming to the required standard The University of British Columbia April, 1990 (c)Satindranath M. Banerjee, 1990 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of / ^ Q T4AJ Y The University of British Columbia Vancouver, Canada Date jfcfo /i/ /ffO DE-6 (2/88) 11 GENERAL ABSTRACT The papers in this thesis represent a series of attempts — empirical and theoretical — to integrate developmental biology with population level studies of variation; to initiate a "developmental population biology" which would complement the well established fields of population ecology and population genetics. The introductory chapter traces the development of the conceptual ideas from the context of the maturation of a single research group. There follow three empirical chapters based on population studies of yellow cedar (Chamaecyparis nootkatensis). The first of these chapters examines the interdependency of progeny growth variation on parentage and stand structure and argues that parentage, developmental history and environmental contingencies can interact in complex ways to structure the variation observed in natural stands. The second chapter examines time related changes in patterns of variation for mainstem growth and needle initiation data of seedlings, and finds that the majority of the increase in variation with time results from differentiation among individual seedlings. The third chapter examines the nature of intra-individual variation in needle (from seedlings) and scale (from mature trees) data from the perspective of the concept of morphological integration, the amount and structure of covariation within an individual. The results of this chapter demonstrate that the nature of morphological integration changes during the course of development, and that variation in morphological integration — that is the pattern of variable relationships or covariance structure — distinguishes individuals. I l l The final chapter is more theoretically oriented, and demonstrates how the patterns of increasing variation with time, and changing covariation with development (Chapters 2, 3) may be unified and explained in the context of developmental trajectories, where such trajectories represent the development of the form of individual organs through time in terms of point trajectories through a multivariate space. The nature of such developmental trajectories is ultimately a manifestation of cell division and elongation in various planes, resulting in the external form of the organs. Three increasingly complex graphical models of developmental trajectories are presented and it is argued that when developmental trajectories diverge from each other in a nonlinear manner, changes can occur in both correlation and covariance structures, coincident with changes in size. The relation between developmental trajectories and the production of variation within populations is further elaborated from the context of dynamical systems theory. iv TABLE OF CONTENTS GENERAL ABSTRACT ii LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix PREFACE, GENERAL INTRODUCTION AND CONTEXT 1 Introduction 3 A Br ief History 6 A Note On Analyt ica l Style 16 References 19 CHAPTER 1: THE INNATE COMPLEXITY OF WITHIN POPULATION VARIATION 21 Abstract 22 Introduction 24 Materials and Methods 25 Results. 27 Discussion 29 Conclusion 33 References 35 CHAPTER 2: VARIATION IN MAINSTEM GROWTH AND NEEDLE INITIATION AMONG AND WITHIN OPEN-POLLINATED FAMILIES OF YELLOW CEDAR 44 Abstract 45 Introduction 46 Materials and Methods 48 Results 50 Discussion 55 Conclusion 62 References 63 CHAPTER 3: A PILOT STUDY OF THE NATURE OF MORPHOLOGICAL INTEGRATION WITHIN AND AMONG INDIVIDUALS 79 Abstract 80 Introduction 81 Materials and Methods 85 Results 91 Discussion 94 References 97 V CHAPTER 4: DEVELOPMENTAL TRAJECTORIES AND THE ONTOGENY OF WITHIN POPULATION VARIATION 108 Abstract 109 Introduction I l l Development as a Complex System 126 References 131 EPILOGUE 141 VI LIST OF TABLES TABLE 2.1 Measurement times for whorl initiation and height data 65 TABLE 2.2 Whorl initiation data: sums of squares apportioned to various putative sources of variation 66 TABLE 2.3 Height growth data: sums of squares apportioned to various putative sources of variation 69 TABLE 3.1 P C A 1 for needle and scale data sets 100 TABLE 3.2 Nested A N O V A for needle data 101 TABLE 3.3 Nested A N O V A for scale data 102 TABLE 3.4 Nested A N O V A for P C A 1 of scale data 103 TABLE 3.5 -ln]R[, THETAISO, and T H E T A P C A values for individual seedlings' needle data 104 TABLE 3.6 -ln|R|, THEATAISO and T H E T A P C A for individual crown thirds based on scale data 105 TABLE 3.7 A N O V A of variation in -ln|R|, THETAISO, and T H E T A P C A within and among open-pollinated families, for needle data 106 vii LIST OF FIGURES FIGURE 1.1 Summary of sampling scheme 37 FIGURE 1.2 Variation between sites represented by notched box-plots 38 FIGURE 1.3 Variation among seedling families within each site as represented by notched box-plots 39 FIGURE 1.4 Variation in seedlings among crown positions for selected trees from the midslope site as represented by notched box-plots 40 FIGURE 1.5 Variation in seedlings among crown positions for selected trees from the swamp site 41 FIGURE 1.6 Diagram plotting out positions of trees relative to an "average canopy" 42 FIGURE 1.7 Plot of germination capacity against canopy position for selected trees 43 FIGURE 2.1 Variation between sites, for seedlings, represented by notched box-plots 72 FIGURE 2.2 Variation among seedlings from O.P. families for the midslope site 73 FIGURE 2.3 Variation among seedlings from O.P. families for the swamp site 74 FIGURE 2.4 The apportionment of variation in seedlings for whorl data 75 FIGURE 2.5 The apportionment of variation in seedlings for height data 76 FIGURE 2.6 Variation among open-pollinated families in whorl data as Eta^ 77 FIGURE 2.7 Variation among open-pollinated families in height data as Eta^ 78 FIGURE 3.1 Variables measured on juvenile needles and mature scales 107 FIGURE 4.1 Two examples of growth curves 134 FIGURE 4.2 The apportionment of variation for height and whorl data 135 v i i i F I G U R E 4.3 Three models of developmental trajectories 136 F I G U R E 4.4 Ellipsoids of variation corresponding to the "pure growth model" 137 F I G U R E 4.5 Ellipsoids of variation corresponding to the "increasing variation model" 138 F I G U R E 4.6 Ellipsoids of variation corresponding to the "increasing complexity model" 139 F I G U R E 4.7 Major axes of ellipsoids of variation from various developmental studies 140 Acknowledgements This thesis owes much to many people. Foremost among them are Jack Maze and Yousry El-kassaby who were mentors, intellectual adversaries of the highest order, and friends who have enriched my life. I owe more to them than I can really express. I am grateful to other committee members -- Gary Bradfield, John Worrall, Wilf Schofield and Don Lester - for their support and encouragement. In particular the editorial skills of Wilf Schofield did much to improve the clarity of the original manuscript, he remains il miglior fabbro. Cy Finnegan is unlikely to accept the responsibility due to him for shaping the philosophical commitments that have guided this thesis. The data gathered in this thesis would not have been possible without the aid of Canadian Pacific Forest Products (C.P.F.P.), the Pacific Forestry Centre (P.F.C.) and the funding provided by a G.R.E.A.T. award from the B.C. Science Council. Both C.P.F.P.'s Growth and Yield Crew, and Seed Orchard Staff risked life, limb, and patience in the course of my field collections. I thank, in no particular order, Clayton Chu, Bill Riel, Stuart Moore, Dennis Ralph, of the growth and yield crew; and Cathy Cook, Debby McLeod, Barb Newberry and Peter Schultz of the seed orchard staff for their patience, help, and friendly abuse. Norm Barnett kindly provided access to the field sites and felled many trees. Yousry El-kassaby and Vlad Korelus of C.P.F.P. must be thanked for allowing me to use their staff. At the P.F.C. I must thank Doug Taylor, George Edwards and Frank Portlock for providing a home for my orphan seedlings, for their continual X kindness in advising and aiding me throughout the time I was using their facilities for germination and growth studies, and afterwards. Among graduate students at UBC, past and present, I had the opportunity to work closely with Kali Robson, Rob Scagel, Peter Sibbald and Len Dyck. Friends first, collaborators second, we have all partaken of that strange well of crazy and critical thought that exists in room 2218, hidden in plain sight behind the "coke bottles". Other students at UBC who offered ideas, companionship, cider, and tolerated me through my more demented moments were Lacy Samuels, Brian Compton, Robin Davidson, Rosemary Mason, Marilyn Cherry, Naomi Checowitz-Beck, Tia Vellani, Rod Campbell, Beth Molitor, Elaine Hooper and Phillipp Funck. Adventures with the Vampire and kidlings gave me a new perspective on developmental complexity, the wonderful things science has nothing to say about, and the painful things scientists momentarily escape from in our solitary craft or art. My approach to science has been oddly shaped by the poets I have known, Mary Lanore, Anne M. Kelly, Robin Skelton, Yvette Brend, Rhonda Roy, the very Poundian, Carl Grindley. Finally, all I have ever wanted to be is a story teller, in any medium, in this medium. My family has always been the source in which I am rooted. When they did not understand what I was up to, they supported me regardless. This thesis is dedicated to the three of them: Mahamaya, Satyendranath, and Sanghamitra. THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR Preface, General Introduction, and Context (or how I learned to stop worrying and love Chaos) Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C. V6T 2B1 2 "What we should do, I suggest, is to give up the idea of ultimate sources of knowledge, and admit that all knowledge is human; that it is mixed with our errors, our prejudices, our dreams, and our hopes; that all we can do is to grope for truth even though it is beyond our reach. We may admit that our groping is often inspired, but we must be on guard against the belief, however deeply felt, that our inspiration carries any authority, divine or otherwise. If we thus admit that there is no authority beyond the reach of criticism to be found within the whole province of our knowledge, however far it may have penetrated into the unknown, then we can retain, without danger, the idea that truth is beyond human authority. And we must retain it. For without this idea there can be no objective standards of inquiry; no criticism of our conjectures; no groping for the unknown; no quest for knowledge." K.R. Popper, 1963 3 Introduction The relationship between data (or observation) and theory is central to the conduct of science, and yet evades simple reduction into some absolute method whose validity is guaranteed. A caricature of scientific methodology may be of two types. The inductive caricature would be as follows, "one gathers data, then by scanning masses of data a theory emerges". The deductive caricature would be, "one has a hypothesis from which one deduces logical consequences and then gathers the relevant data to test those deductions". Obviously, there is a relationship between data and theory, but often the relationship is less than obvious. For example, if in a study of morphological variation among a number of individuals in a particular species it was determined that X% of the variation could be apportioned to individuals being subject to different environments (say different sites), Y% is apportioned to geneological relationships (i.e. the parents of the individuals under study) and Z% is apportioned to developmental differences within an individual; for example variation among sequential structures such as leaves in a plant, or cells within a given tissue in an animal. If the question were posed, based on these data: "which source of variation is of primary importance to evolutionary theory? " Is it that source to which most of the variation was apportioned? Or, are only certain types of variation important for evolution, for example those that can be correlated to survival? The relationship between data and theory often seems akin to that between two (or three) poles. For- example, several long pieces of wood, when placed upright are likely to fall over, but notch each piece, lean them together, suddenly they support a tarp and you have a tent. I am concerned with the relationship between data and theory 4 for a particular reason. I use data to argue that attempts to explain variation among individuals within a population as the consequence and interaction of genetics and environment are incomplete. Based on data, I further develop an argument that major patterns of variation among individuals may be seen as a consequence of epigenetic factors, i.e. they are rooted in variation resulting from development. Thus, again I use data in a dual fashion, to test a hypothesis, and to build one. Induction and deduction leaning against each other. Carrying the analogy further, if I am lucky I might find a tarp to keep the rain out. This study considers the nature of morphological and growth variation in Yellow Cedar seedlings, and their parent-trees originating from two contiguous, but ecologically distinct, sites. At each site, I sampled the morphology of the parent trees, and then gathered seed from these same parent trees to set up a progeny test (of open pollinated families; i.e. they share only the maternal parent) where again I gathered morphological data on early growth and development in the seedlings under common garden (or nursery) conditions. Thus, roughly, I have three sources to which I can apportion variation in these seedlings (or their parents): differences among sites, differences based on genetic differences among individuals (either from maternal parentage or segregation, recombination and an assortment of pollen parents) , differences resulting from development. The structure of the thesis is as follows. Following this introductory chapter there are three empirical chapters. The first is an exploratory analysis of sources of variation in mainstem growth data of progeny from the parent trees on the two sites. The concern is putative sources of complexity - vagaries of the mating system and site topography ~ and the relationship between data structure and model assumptions when 5 quantitative genetic theory is applied to natural populations. In the second chapter I examine time trends in variation for two variables estimating development — mainstem growth, and whorl initiation — and find that the strongest trend is one of increasing variation among individuals within a family. These results are used to relate genetic and epigenetic sources of variation to perceived patterns emerging with development. In the third chapter, I explore the nature of within individual variation (morphological integration) in parent-trees and their progeny. The focus is on how the production of sequential structures (e.g. leaves) can impact on intra-individual variation among those structures and the differences in the organization of that variation between juvenile needles and mature scales. In a concluding chapter the material from the three empirical chapters is integrated with data from other projects I have been involved in over the last few years to develop a heuristic model of the relation between development of individuals and patterns of variation within a population. While my approach is primarily graphical, the models I present may be formalized into explicit mathematical models and suggest new protocols for quantifying data gathered on developing and evolving systems. Finally, there is an epilogue for foresters where I emphasize what I believe to be the practical consequences for applied forestry of some of my results concerning the nature and sources of variation within populations. 6 A Brief History In this, the prefatory chapter, I take a more personal approach and briefly sketch the history of the research program that led to this thesis. One of the main themes in this thesis is that individual development is a major source of the variation found within populations, so it is fitting that I begin with a brief history of the development of the research program which led to this view. While intermingling scientific and personal history might seem an unusual method of introducing the concepts of an academic thesis, I believe it represents an honest and accurate appraisal of how the ideas I discuss in this thesis came about. Science is a human affair, conducted by and for other humans, and it seems presumptuous to try and remove the human element from science under the guise of "objectivity". What objectivity there exists in science does not result from the rhetorical stance of scientists but is related to the fact that reality exists independent of our individual efforts to try and comprehend it. If I need philosophical justification for my approach, I would note that my mixture of conceptual and personal narrative stems from an approach to understanding science with such eminent proponents as Arthur Koestler (1959: The Sleepwalkers), Paul Feyerabend (1975: Against Method) and David Hull (1988: Science as a Process). Admittedly, even in the absence of historical precedence I would have written the introduction much as I have and offered no apologies. To misquote Popper, all knowledge is human, truth is not. The approach taken in this study is the consequence of a particular research program in which subject matter is wide ranging (from DNA, to embryology, to population studies, to systematics) but unified by a common theme emphasizing variation. Our focus is the amount of variation, its 7 structure, and how it changes with time in plant morphology. Later, I will elaborate on the significance of "amount" and "structure" of variation, but for now, I wish to briefly, and anecdotally trace the history of the research group. The impetus of this research program was a controversial new theory of evolution largely developed by a (then) young assistant professor at UBC, Dan Brooks (Wiley and Brooks, 1982; Brooks and Wiley, 1986) which argued that the increasing complexity which characterizes organismic evolution could be seen as a manifestation of a natural law, the second law of thermodynamics. Essentially Brooks and Wiley argued that "structural" complexity — the development of biological forms ~ could increase in a manner analogous to the entropy of strictly thermodynamic systems. In short, evolution was the consequence of, indeed driven by natural laws, and hence would occur in the absence of selection which was reduced to the level of a contingent force (i.e. sufficient but not necessary). In formulating this approach, Brooks and Wiley made a major break with previous "thermodynamic" views of evolution by identifying structural complexity (the number of interacting parts that compose a system, and the nature of their interactions) rather than physical disorder (classically related to simply the number of configurations the parts of a system can obtain, usually ignoring interactions by assuming the parts to be independent; e.g. gas in a box) as increasing with time in a law-like manner. Furthermore, by emphasizing time related changes in structural complexity, they explicitly invoked a strong role for developmental phenomena in evolution. The theory, while provocative, was also troubling, the common reaction was "yes, these ideas are interesting, but are they testable? Can they be 8 used?" Some of its core terms, "entropy", "order", "organization", and "complexity" are extremely hard to define. It was reasonable for a biologist to ask of the theory, "What does this have to do with biology"? At the time that Dan Brooks was developing these ideas, two of his key sounding boards at the University of B.C. were Jack Maze and Cy Finnegan. At that time Jack Maze was primarily an embryologist and morphologist whose research interests periodically oscillate between studying grass embryology and population level studies of vegetative and reproductive morphology in conifers. Cy Finnegan was Dean of Sciences (at the time), a mature scientist whose research program was concerned with the development of neural crests in axolotls, but who was now primarily involved in administration, with the exception of his teaching a course on the history and philosophy of biology. Though Finnegan did not directly participate in the elaboration of the theory, his influence was considerable. Many ideas were filtered through him and he provided a conceptual and historical context for some of the basic concepts underlying Brooks and Wiley's work. In retrospect, Finnegan provided an atmosphere in which a new theory could grow. At this time Maze had recently acquired a new graduate student, Rob Scagel who complemented Jack's interest in the complexities of morphology with a strongly analytical and operationalist frame of mind; theories as intellectual or semantic play-toys did not interest him. Rob had considerable experience in applied forestry, and was given to using empirical applicability as an Occam's razor by which to test ideas and concepts. Thus, as Brooks and his collaborators set out to develop their ideas into an elaborate theory, Scagel and Maze focused on how, or if, any of these ideas could be applied to 9 the data sets commonly gathered by developmental biologists and evolutionary morphologists. While Brooks and Wiley first developed these ideas in the context of phylogenetic systematics, their area of expertise, Scagel and Maze concentrated on another level of biology; variation among individuals and populations. In testing these ideas, Scagel and Maze focused on what they considered to be the core statements of Brooks and Wiley's theory. First, the theory stressed "entropic increases in biological systems". Biologically, this was interpreted to mean that if one were to examine a developing system, it would become more complex with time. This increase in complexity should be reflected in increasing variation and changing variable inter-correlations. Secondly, Brooks and Wiley's theory implied that the production of variability plus historical and developmental constraints were in themselves sufficient and necessary for evolution to occur, an implication relegating selection to a "rate control", rather than a driving force of evolution. Scagel and Maze saw this as implying, at the population level, that variation within populations (resulting from production of variants, a form of entropic increase) should be greater than variation among populations in different environments. Thus, at this point (circa 1983) there existed two research programs running in tandem. One was Brooks and Wiley's which was attempting to extend, elaborate and formalize its central metaphor, "evolution is an entropic process" into a full blown theory of evolution co-opting development, population genetics and ecology into a new synthesis based on concepts from information theory and nonequilibrium thermodynamics. Simultaneously, Scagel and Maze were ignoring the elaborations of the theory to concentrate on the empirical application of its central analogy, 10 that is they identified "entropy" with variation in biological structures. In consequence, they focused on methods of quantifying different aspects of these phenomena, emphasizing two items. The first item of concern was the nature of changes in variation with time in developing systems. To examine such changes Maze and Scagel initiated a series of embryological studies examining how anatomical variables change with time in both their variability and inter-relationships with other variables. The second item of concern was the nature of variation within and among nested subsets of the biological hierarchy (i.e. individuals/populations/species), which was investigated via morphological studies of variation with-in and among populations and species to see at what hierarchical level the variation apportioned to. In summary, one group (Brooks and Wiley) was proceeding deductively, while the other group (Maze and Scagel) was proceeding inductively, for the most part. What is unusual is that although these two research programs were running in tandem, were based on elucidating a common metaphor, were located in the same university, and whose primary instigators, Maze and Brooks were friends, there was remarkably little formal collaboration. Ideas were discussed in Dr. Finnegan's office while stealing his coffee and gossiping about colleagues, and then each investigator went his own way. Maze and Scagels's work received no mention in the first edition of Brooks and Wiley's book on their theory, although it was included in their second edition. Similarly, though Maze was the first to support the theory in print (Maze and Bradfield, 1981) and most of Maze's papers after 1981 mention Brooks and Wiley's theory in a favorable way, none of the formalisms based on information theory are utilized (until 1989), nor are any of the specific models that Brooks and Wiley elaborate for development or populations 11 utilized in the empirical studies on populations and development of Maze and Scagel. Thus we have the remarkable juxtaposition of two research groups that are conceptually inter-twined but are virtually methodologically independent in the way they choose to elaborate and apply those concepts; remarkable little cross-fertilization was occurring in the hallways of UBC. While a sociological analysis such as Hull's (1988) might provide insight into the group dynamics, I wish to stress a methodological problem, before returning to the narrative. At some point a theory must connect with data. Most of Brooks and Wiley's ideas were stated in terms of formal concepts from information theory and were applied to illustrative (usually artificially generated) examples. However the formalisms (for example, variants of the Shannon-Weiner equation) of information theory, based on the distribution of discrete characters, were not particularly amenable to realistic data sets that could be gathered on metric characters. Given this, Scagel and Maze based their analysis on protocols from the field of multivariate morphometries (Pimental, 1979; Bookstein et al., 1985) , dealing with variable dispersions and covariances. They then sought to interpret patterns found in the course of morphometric analysis in terms of concepts from information theory, noting that more information would be required to describe a developing structure/evolving population, as the total amount of its variation increased, or if the structure of that variation changed with time. In trying to clarify the relationship between their empirical results and concepts from information theory, Maze and Scagel were aided by John Collier, a philosopher specializing in information theory. He was also an early (though not uncritical) defender of Brooks and Wiley's (Collier, 1986) use of information theory, and thus provided a filter through which both groups 12 ran their interpretations of biological phenomena in information-theoretic terms. It was only recently our group came to understand how these intuitive explanations could be formally related to the mathematical quantities of information theory (Banerjee et al., 1989). At this point, Maze received another graduate student, Kali Robson. Her particular areas of interest were systematics and beer, and she had a profound respect for both. As she became familiar with Maze and Scagel's work, she became interested in how their results concerning developmental variation and variation in populations could be related to the patterns she found in between species variation. Her interest was in connecting these two elements, individual development and population variation to differences among and within species. Thus, she was interested in discerning causal relations between different tiers of the biological hierarchy. Her particular interest was in what Eldredge and Salthe (1984) termed the "geneological hierarchy" in which individuals, populations, species, can be seen to form nested subsets, where higher levels involve more inclusive subsets. Furthermore, as one ascends the geneological hierarchy, one is positing greater distance to the nearest common ancestor for any given pair of taxa. Until recently, most explanations linking individual, to population to species level patterns have been given in the context of genetics, and interactions between genotypes, the phenotypes they determine and the environment; i.e. explanations were given primarily in the context of selection and fitness. However, recently, numerous authors have pointed out that such selective explanations are incomplete because they ignore the role of developmental constraints on form (Gould, 1980, 1989; Maynard-Smith et al., 1985). In light of these caveats, the emphasis of the studies conducted 13 was on morphological variation from the point of view of developmental processes leading to the observed form. Thus, at this point there were studies at three levels of the biological hierarchy: studies of variation among individuals, variation among populations, and interspecific differences. Slowly the relationship between these levels began to be seen from the point of view of an interaction between processes inherent in ontogeny and phylogeny in that development concerns those events that lead to the form manifested by an individual while evolution encompasses those events which explain the ontogenies expressed in a lineage. Kali's M.Sc. thesis focused on the relationship between the structure of within species variation, from the perspective of cladistic studies of among species variation; showing that with speciation there is a change in the structure of within species variation which is unpredictable. In her Ph.D (at the University of Nebraska) she continued these studies, but introduced explicit developmental and environmental components, studying in situ for two years the developmental patterns of a series of species in the Balsamorhiza complex. In this work she found that certain variable relationships characterize different levels of a hierarchy, i.e. they are historically constrained. In both theses Kali juxtaposed the results of analyses of intra-specific variation against an independently derived phylogenetic hypothesis. What this permitted was the interpretation of the patterns of developmental and intra- and inter-population variation in the context of an estimate of phylogenetic relationships among species. Again, we have two tiers leaning against each other I entered the research program at the time that Kali was leaving, and we had spent much time discussing the ways in which variation at various 14 putative levels might be connected. These discussions left me convinced that I wanted to conduct a study analogous to Kali's, that is one linking various levels of the biological hierarchy. However, at the time, I was not clearly aware how this could be accomplished. It was this, plus my previous training as a forest geneticist, that caused me to focus on the nature of variation within a population; it is this variation which the forest geneticist seeks to manipulate. It seemed to me, that there were three sources of such variation — developmental, environmental, and genetic. I hoped to collect data sets from each source, and then correlate the different data sets. In the course of both data gathering and analysis, I gradually began to understand the logistic difficulties of such a scheme. Thus, instead of my original, overly ambitious plan of apportioning variation as a result of X% for each source in some sort of absolute manner, I focused on the effects of the single source of variation usually ignored in population studies, that caused by development, and sought to determine to what effect it could lead to major structure in the data. As a corollary to my study of development, I also gathered some modest data which would allow me to place my developmental results in a wider context, gathering data on the positions, and canopy locations of parent trees in the two collection sites and conducting my developmental studies in the context of an open-pollinated progeny test. Within my first year of graduate studies another member was added to our research group, Peter Sibbald. He had actually entered graduate studies at UBC botany at the same time as Kali Robson, and his official field of interest was photosynthesis research from a molecular biological perspective. However, his unofficial fields of interest included just about everything, and thus he was often a very good sounding board for other graduate students to 15 test out new ideas, since he was capable of doing a remarkable imitation of Occam's razor. As his interests turned to more theoretically oriented issues in molecular biology, and for reasons beyond the scope of this essay, he decided to switch supervisors and "officially" joined our research group. While other members of our group were interested in variation at the organismal level, Peter dealt with variation in DNA sequences. In the course of his work he introduced a strongly deductive component to our research. Peter's method of choice is to begin with an assertion (either reasonable or ridiculous), derive a consequence, and then to gather data. Based on the data he would modify the assertion, again derive a consequence and so on. In his thesis Peter used, and developed a number of new mathematical techniques for data analysis, and became very familiar with the mathematics of information theory, its relationship to statistical inference, and its limitations in data analytic operations. Peter, Jack and I spent much time discussing whether the theory that was elaborated by Brooks and Wiley was indeed testable. For some time, our research group was increasingly aware that many of our results conflicted with certain of Brooks and Wiley's speculations, however since the methods we utilized were very different from the methods of Brooks and Wiley , it was not at all clear at first how we could bring our data and their theory into apposition. Such considerations raised the question, what forms of mathematical models would be consistent with the results of the data we were collecting, and could those mathematical models be restated in terms of the formalisms from information theory used by Brooks and Wiley? Secondly, could concepts from those areas of inquiry suggest new ways of analyzing the data. It was at this point we started exploring the possibility of linking the analyses we conduct more formally to information theory. At 16 the same time, Peter, Len Dyck (an independent but kindred spirit) and I began exploring concepts from fractal geometry and nonlinear dynamical systems theory. Our original motivation was recreation and interest, and the feeling that many of the problems we were examining in biological data sets, were a consequence of the nonlinearity of biological data, leading to complex forms of behavior best characterized by concepts from nonlinear dynamical systems theory. Eventually these separate sets of ideas and interests coalesced into a publication (Banerjee et al., 1989) where morphometric analysis was formally related to formalisms from information theory in such a way that the Brooks-Wiley models could be tested, and concepts from nonlinear dynamical systems were used to (a) suggest extensions to the protocols developed, (b) develop the notion that the changes in variation and correlation structure we had previously encountered in developmental studies could be modelled in terms of diverging nonlinear developmental trajectories. The notion of viewing the development of an organ as a trajectory through the space of the variables measured, and its consequences is taken up in the final chapter of this thesis. A note on analytical style. Throughout the empirical chapters in this thesis, the analyses done are in the context of exploratory data analyses. It seems to me that if I wish to examine the relevance of a particular theory, I must, at least initially, use analyses that were derived independent of that theory to avoid circular reasoning. One of my aims, in using the analytical protocols I have chosen, is to analyze for pattern while making as few process assumptions as 17 possible. For instance, I do not calculate heritabilities, but calculate the variation attributable to the fact that we have different parentage. I do this, because to calculate heritability, one must assume a particular mating scheme and a particular model of genetic transmission. For open pollinated families heritability is often calculated under the assumption that we are dealing with a half-sib mating structure. If we have a strong basis to believe these assumptions hold, then it is appropriate to calculate heritability, since the methodology is a logical deduction based on those process assumptions (Falconer, 1981; Jacquard, 1983). However, if we are not as sure of the mating design, or of the actual genetics underlying the characters we measure, we should choose a weaker set of assumptions to guide our analyses of pattern (Banerjee and Maze, 1988). Similarly, I express variation in terms of ratios of sums of squares rather than in terms of variance components. There are two reasons. First, variance components are derived from the original sums of squares. Secondly, they require stronger assumptions about equality of sample size and heterogeneity of variances. In my personal experience, for "well behaved" data sets, sums of squares and variance components tend to give essentially the same apportionment (see Maze et al., 1989). Naturally, one must make some assumptions. For example I examine the properties of determinants of correlation matrixes. Such a quantity is really meaningful only if the data structure approximates multivariate normality. However, it is possible to test that the assumed data structure is indeed approximated by the data gathered. Similarly, in genetically well defined studies, it is easier to test if the assumptions invoked in a quantitative genetic analysis hold. Sober, (1988) provides a useful description of the role of parsimony in biological data analysis in 18 terms of "the epistemological problem of pattern and process" (see Sober pg 11-12) where he argues that, for systematics, "The data in and of themselves do not permit us to conclude that one geneology is true and all alternatives are false. Rather we hope that minimal and plausible assumptions about process will allow us to use the data to discriminate among competing phylogenetic hypotheses. Just how minimal such assumptions can be and still do the work we expect of them ~ of forging an evidential connection between characters and geneology — is the central problem in the foundations of phylogenetic inference" Needless to say, the problem is actually central to all cases of deductive inference and may be restated as follows: given a data set, what minimal assumptions do we need to make about the data before we can begin to analyze it and derive patterns from it; i.e. what is the minimal degree of circularity in our explanations that we are willing to live with?. How do we obtain knowledge in spite of what we think we know? 19 References Banerjee, S. and J. Maze. 1988. Variation in growth within and among families of Dowlas-fir through a single season. Can. J. Bot. 66: 2452-2458 Banerjee, S., Sibbald, P.R. and Maze, J. 1990. Quantifying the Dynamics of Order and Organization in Biological Systems. J. Theor. Biol. 143:91-112 Bookstein, F., B. Chernoff, R. Elder, J. Humphries, G. Smith and R. Strauss. 1985. Morphometries in Evolutionary Biology. Special Pub. 15, Acad, of Nat. Sci of Philadelphia. Brooks, D.R. and Wiley, E.O. 1988. Evolution as Entropy, 2nd ed. University of Chicago Press, Chicago. Collier, J. 1986. Entropy in Evolution. Biol and Philos. 1:5-24 Eldredge, N. and S.N. Salthe. 1984. Hierarchy and Evolution. In R. Dawkins and M. Ridley, eds., Oxford Surveys in Evolutionary Biology. 1:182-206 Falconer, D.S. 1981. Introduction to quantitative genetics. Longman Press, London. Gould. S.J. 1980. The Evolutionary Biology of Constraint. Daedalus 109:39-52 Gould, S.J. 1989. A Developmental Constraint In Cerion, with comments on the definition and interpretation of constraint in evolution. Evolution 43:516-539 Hull, D.L. 1988. Science as a Process. An Evolutionary Account of the Social and Conceptual Development of Science. The University of Chicago Press. Chicago. Jacquard, A. 1983. Heritability: one word, three concepts. Biometrics 39: 465 -477 Koestler, A. 1959. The Sleepwalkers. Hutchinson (Reprinted, 1970, PEnguin Books Ltd. Harmondsworth, Middlesex, England. Maynard-Smith, J., Burian, R. Kauffman, S., Alberch, P., Campbell, J., Goodwin, B., Lande, R., Raup, D., and Wolpert, L. 1985. Developmental Constraints and Evolution. Quart. Rev. Biol. 60:265-287. Maze, J. and G. Bradfield. 1981. Neo-Darwinian Evolution --Panacea or Popgun. Sys. Zoo. 31:92-94 20 Pimental, R.A. 1979. Morphometries: the Multivariate Analysis of Biological Data. Kendall/Hunt Publishing Co. Dubuque, IA. Sober, E. 1988. Reconstructing the Past. Parsimony, Evolution and Inference. The MIT Press. Cambridge, Mass. Wiley, E.O and D.R. Brooks. 1982. Victims of History — A Nonequilibrium Approach to Evolution. Sys. Zoo. 31:1-24. 21 T H E O N T O G E N Y O F M O R P H O L O G I C A L V A R I A T I O N : A N E X A M P L E F R O M Y E L L O W - C E D A R Chapter 1: The Innate Complexity of Within Population Variation Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C., V6T 2B1 22 Abstract Variation encountered in developing plant structures is usually attributed to and examined with respect to three sources: (a) parentage, (b) developmental history and (c) environmental contingencies. This chapter examines variation in early seedling growth for open-pollinated progeny of yellow-cedar as a consequence of items (a) and (c), specifically examining the inter-dependency of progeny variation on parentage and stand structure. The seedlings in this analysis were collected from different thirds (upper, middle, lower) of the crowns of individuals of C. nootkatensis occupying two ecologically distinct but contiguous sites. Variation in seedling height growth between sites, among parent trees within a site and among crown sections within a single tree were assessed using notched box-plots and ANOVAs. Patterns of variation in the progeny were complex but appear to be related to position of parent-tree crowns relative to the canopy and resulting pollen clouds. The degree of with-in crown variation was of equal magnitude or greater than that between crowns and was variable from tree to tree. This with-in crown variation in progeny was examined in the context of parent tree locations and size. The results are discussed in relation to the inter-dependencies between patterns of growth variation, mating system and stand spatial structure as well as their consequences for quantitative genetic studies of natural populations. Specifically, canopy size and topography locate individual canopies relative to each other. The degree to which individual tree crowns overlap with that of their neighbors will affect the 23 likelihood of outcrossing events. Trees whose crowns are isolated from their neighbors, either horizontally or vertically are more likely to have selfed progeny. Thus, with-in an uneven-aged stand there is potentially a continuum of mating-systems from highly out-crossed, to highly selfed, even with-in the crown of a single tree. These lead to severe violations of the assumptions of half-sib relations among progeny used in quantitative genetic studies on open-pollinated progeny and suggest natural stand seed collections for reforestation should concentrate on collecting from the mid-crown and from co-dominant trees so that the resulting progeny are maximally likely to be the product of out-crossing events. 24 Introduction Population level studies on conifers are notable for the degree of variation encountered within as opposed to among populations for variables assessing growth (Khalil, 1987; Magnussen and Yeatman, 1987a,b,). This high with-in population variation is due either to crossing between intrinsically variable individuals or as a consequence of the mating system, where progeny represent a mixture of half-sib families resulting from open pollination. (Banerjee and Maze, 1989, Maze et. al, 1989). Indeed, open-pollinated families from natural populations also display a large-degree of with-in group variation (Harry, 1987; Scagel et al, 1987; Wells and Toliver, 1987). Maze and Banerjee, 1989 demonstrated that the variation with-in full-sib families was of comparable magnitude to a heterogeneous mix of crosses thus indicating the high variation due to crossing between intrinsically variable parents. However, in natural stands, in addition to the intrinsic variability of the parent trees, other factors must be considered such as the complexity of the mating system and its ability to structure and either promote (panmixis, outcrossing) or constrain (selfing, effective selfing, consanguineous mating) this innate variability. In yellow-cedar where stands are uneven aged and spatially structured (Antos and Zobel, 1986), the genetic variation in the progeny may be contingent on a number of factors such as the intrinsic variability of the parent trees, the degree to which neighbouring trees are related, and the position of individual tree canopies relative to that of neighbouring trees. In this study I examine this with-in population variation at a finer scale, by examining variation with-in sub-sections of individual tree crowns for trees 25 whose sizes and positions in the landscape were mapped. The questions I addressed were, among others, the following: (1) What is the magnitude of progeny differentiation with-in a canopy? (2) Can differences between progeny from different sections of a tree be related to ecological factors such as tree size, the topographic complexity of the landscape, and the degree of isolation of cones within a canopy? In essence, I was interested in the degree to which local contingencies affect the growth of resulting progeny, and how this may affect what is percieved as the quantitative genetic variation. Specifically, there is a need to decompose the often large error term in Forest Genetic models for examining variation in O.P. families. Due to the expense of controlled crosses in conifers, initial estimates of heritable variation are often based on natural stand seed collections. These results may have operational consequences in the planning of natural stand collections. While subsequent chapters examine population variation from the perspective of development and its interaction with genetics, this chapter seeks to present a descriptive account of the degree to which ecological factors such as canopy structure may also interact with genetic, developmental (and physiological) factors to result in complex patterns of fine scaled variation in natural populations. Materials and methods Trees were sampled from two contiguous, but ecologically distinct sites on Walker Mtn. in Sooke region, Vancouver Island (Figure 1): a swamp and mid-slope site. A total of 57 parent-trees were felled and sampled across the two sites (26 in swamp, 31 midslope). Seed were collected and individually 26 bagged from the upper, middle and lower crown thirds of each tree. Of the 57 trees, sufficient seed was obtained from 20 (9 swamp, 11 midslope) to initiate a progeny trial. Not all thirds were represented through lack of cone production. The 20 trees were represented by 35 samples of thirds with 30 seedlings set out for each sample. The design consisted of five randomized replications of 6 seedling rows per canopy sample set out in "4-15" styroblocks. To avoid edge effects there was an edge row which was not included in analyses. The seedlings to be discussed were measured for height twice, once in January (4 months after germination) and once in June (9 months after germination) of 1988. Germination data were also gathered. Seed were winnowed until they were approximately 80% filled seed. From the winnowed seed percent germination data were gathered for two 100 seed samples for each crown third represented in the trial. The average of these two samples was graphed for a number of selected trees against crown third. The germination procedures followed standard ISTA (International Seed Testing Association) protocols. The primary analytical techniques for the growth data were notched box-plots and one-way ANOVAs. Notched box plots were introduced by exploratory data analysts (Mcgill et al., 1978; Chambers et al., 1983) as a graphic summary of variation within and between groups. Specifically, these techniques graphically assess the median, quartiles, outliers and confidence limits about the median in any given sample (often called a batch) of data. Box plots are useful for simultaneously comparing the amount and "shape" of variation within groups and the differences in these quantities between groups. The "notches" represent the confidence limits about the median and if they do not overlap, then the medians may be 27 considered "significantly different" with 95% confidence. Box plots and simple one-way A N O V A s were conducted to assess variation between sites, between trees for each site and between crown thirds for a given tree. While it is possible to combine all the analyses done into a single fully saturated nested model, I wished to explore the patterns of variation on a per site and per-tree basis, which would have been obscured on such a model. One-way A N O V A s as used here represent a simple and robust analysis without imposing major transformations on the data. Examination of pattern was emphasized in this, the exploratory state, of analysis, therefore the data was retained as raw as possible, rather than imposing structure; my intent at this stage of analysis was exploratory and in the context of hypothesis generation rather than confirmatory analysis. Results Figure Two presents box-plots, illustrating variation with-in and between sites for the two times. The top-right hand corner represents the percent variation in the data — as a ratio of sums of squares from a one way A N O V A — accounted for by sites. There is little differentiation between progeny from the two sites at each time. From times one to two there is an increase in variability and a switch in median rankings, implying a slight shift in relative growth rates from time one to two between the two sites. Variation between trees in each site are presented in Figure Three. For both times and sites, the between tree patterns are similar in generalities. With-in a site, there is somewhat more variability than between sites. With time, variability in the progeny has increased and rankings have shifted somewhat. Open-pollinated families differ not only in terms of median 28 positions, but also in terms of variation about the median. While the notches of most box-plots overlap, certain trees are significantly different in median progeny height. There seems to be a greater number of shifts in relative rankings among midslope trees from time one to time two than among the swamp trees. Again, a shift in median position for an open pollinated family (for example mid-slope tree #3) from time-one to time-two implies a shift in growth-rate relative to other families, and a change in the shape of variation of an open-pollinated family between times one to two implies a change in the degree (increase or decrease in variation) and direction (changes in skew) to which individuals with-in the family are becoming different from each other. From each site three trees were examined in more detail for with-in crown variation. The trees were chosen to illustrate the range of patterns among crown sections and across times, from trees that maintained consistent patterns across time to those in which rankings among crown sections changed with time. The results are summarized in Figures 4 (midslope) and 5 (swamp). Even a cursory scan of these two figures reveals a large degree of complexity: no single pattern is consistent from tree to tree. In some trees (e.g Midslope trees #7 and #11 at time two ) there are significant differences between median progeny heights from different thirds. In others, progeny from different thirds are similar in both median position and variability (e.g swamp tree #4). The amount of variation in a tree accounted for by the thirds ranges from about 0 to 15%. While there is a general pattern of decreasing height with decreasing canopy thirds, it is not consistent. In some cases, like midslope tree 3 at time 1 ~ the progeny from the middle of the canopy have greater average growth than either the top or bottom thirds. 29 Figure 7 presents the germination capacity data for thirds for selected trees in the study. There is a general trend, for both sites, of decreasing percent germination from the top downward in a canopy, but patterns are inconsistent from tree to tree. Note also, that the results for the swamp site, which had a more even topography, are much less complex than for the midslope site in that there appears to be a consistent decrease in germination capacity from the top to the middle canopy third. D i s c u s s i o n The degree of complexity in these results, in terms of heterogeneous variances among open-pollinated families and both heterogeneous variances and median differences among progeny from different thirds of a single tree, violates the distributional assumptions of highly structured/saturated models based on quantitative-genetics theory. Open-pollinated families are often treated as the product of a half-sib mating design where the theoretical correlation between half-sibs is 1/4 of the additive genetic variation, usually calculated by a variance components approach (see El-Kassaby, 1987 for details on calculations and Maze et. al, 1988 for an alternative apportionment based on ratios of sums of squares). Such additive genetic models implicitly assume the following: regular diploid behavior, randomly distributed genotypes, panmixis, lack of epistasis and maternal effects (Scagel et al, 1987). If such assumptions held then there should be no differentiation between canopy thirds and their variances should be equal. Obviously this is not the case. This does not mean such models are completely useless, only that supplementary information, outside the scope of the model, is needed to aid interpretation. The natural situation is much 30 more complex than the theoretical premises used in constructing such models. Quantitative genetic models often are based upon strong assumptions that impose structure on the data (see, for example, Ayres and Thomas' (1990) discussion on the effects of the effects of different variants of the mixed-model ANOVA on the calculation and interpretation of mean squares), and hence are more appropriately used as conceptual guides rather than analytical tools. As Ayres and Thomas note (1990:225): "Careful graphical analysis may be the best safeguard against erroneous biological conclusions". The large degree of residual variation (that not accounted for by the model) within families in progeny trials on natural populations in conifers has been suggested as due to the spatiotemporal variation that exists in the canopy (Fowler, 1965a,b). Scagel et al. (1987) in a study of open-pollinated families of Douglas-fir found such residual variation to be highly structured, suggesting it is not simply experimental error. This led them to suggest that: "Progeny trials should take into account within-parent sources of variation to provide a precise estimate of differences between families and the results of any imposed or hypothesized selection regime. Furthermore the result of such a study would, by necessity, be population specific and would be conditional on the age and architectural complexity of the parent trees, the genetic structure of the stand, and the effective matings. Such an approach takes into account both extrinsic and intrinsic causal mechanisms as opposed to the traditional approach which has been primarily extrinsic". (Scagel et al, 1987:pl620) Thus, to understand what is happening, we must address the specifics of individual trees: how they were growing back on site, and their crown positions relative to the overall canopy. Figure six represents the parent trees studied on the two sites in terms of their height and canopy lengths. The two dotted lines represent average 31 total height and average height to canopy for the trees studied. This provides a preliminary index of crown position relative to the total canopy (of cone bearing trees), delineating a "window" where we might expect the likelihood of outcrossing to be maximal (actually, since female cones aggregate on healthier branches at the top of the tree and male cones on weaker branches at the bottom (Owens and Molder, 1984), such a "window" is somewhat biased) The topography of the midslope site makes the situation even more complex than it appears in figure six. Comparing these figures and relating them back to the previous results on progeny growth, provides some clues towards explaining the array of responses in with-in crown differentiation. Trees whose progeny had a regular decrease in median height with crown third are smaller. Midslope tree #3, which had a higher value for the middle, rather than the top-third, has a crown that extends quite beyond the main canopy. Trees of intermediate size would have crowns with the greatest degree of overlap with the crowns of neighbouring trees. From this set of scenarios it should be clear that the position of individual trees relative to each other, and the associated pollen dynamics can lead to divergent mating systems in different sections of the canopy (Shea, 1987;El-kassaby and Davidson, 1990; Omi and Adams, 1986) . The germination results might seem to support such an interpretation. With degree of inbreeding, one might expect more embryonic lethals and hence, a lower germination capacity . However, there might also be a developmental/physiological component to the germination data that may complicate the population genetic scenario. As previously mentioned, female cones tend to occur more abundantly on the healthier branches at the top of the canopy, and male cones on weaker branches at the bottom. Thus, when 32 female cones occur lower in the canopy, where less light penetrates, there is the possibility they receive less energy (hence, less nutrients) than those at the top, hindering megagametophyte development, and the production of reserves in the megagametophyte. Controlled crosses at various locations within a canopy (or under various degrees of shading) would be required to examine the relative contributions of parental and nutritional effects (within a parent) in determining germination capacity and germination rate. These results can be put in the context of expectations from mating-system theory. Suppose variation among crown sections is due to the degree of changes in mating system, in different sections of the canopy. Then, we might expect the likelihood of selfing will be greater in crown regions above and below an "average" crown or "window" based on neighbouring trees. From this perspective we can visualize, with-in the crown of single trees, a continuum of mating systems from largely outcrossed to largely selfed. The nature of such a continuum will depend upon both crown-size of the individual trees and their topographic locations relative to each other. Thus if (a) the likelihood of selfed progeny is due to crown position and (b) selfing leads to inbreeding depression which may be manifested as (i) embryonic lethal abortions and decreases in germination capacity ( Griffin and Lindgren, 1985; Woods and Heaman, 1989), and (ii) a reduced growth rate among surviving progeny, then there should be a negative relation between the % overlap of individual tree crowns with the average tree crown and the amount of variation among canopy sections . Furthermore, there should be a positive relation between % overlap of any single canopy section with the "average canopy" and the mean height or germination percentage. 33 Besides selfing due to vertical crown position, there is the possibility that neighbouring trees may be related, or if due to layering, genetically identical. Antos and Zobel (1986) have documented the frequency of such layering and it was observed on the midslope site in this study. Furthermore, inbreeding affects not affect only median values via inbreeding depression but also variability. Theoretical expectation is that, relative to total panmixis, low levels of inbreeding will lead to more variable offspring as "hidden" recessive become expressed, while at higher levels of inbreeding the variation decreases due to allelic fixation. Empirical results have been somewhat equivocal on this matter. Sniezko and Zobel (1988) found, for loblolly pine, a positive relationship between variation for both height and diameter growth with degree of inbreeding. However Strauss (1987) examining the relationship between heterozygosity and variability of trunk growth for inbred and crossbred families in Pinus attenuata found no strong trends, except when the data was polarized among inbred and cross-bred families. Still, from the perspective of this study, the menagerie of box-plot sizes and shapes - estimators of variation - both among open-pollinated families and with-in progeny from a single tree is suggestive. Conclusion The results presented in this chapter suggest a complex pattern of with-in crown progeny variation in two natural stands of yellow cedar and I have attempted to decipher some of the complexity as a function of the spatial structure of the canopy in unevenly aged and topographically complex 34 stands. This might provide some insight in explaining the difficulty experienced in nurseries with germinating and growing yellow cedar. It also suggests, somewhat counter-intuitively that natural stand collections should focus on co-dominant as opposed to dominant trees in the stands and limit collection to the upper thirds, or halves, of the crown . The co-dominants are more likely to have their offspring resulting from out-crossing events and thus less subject to inbreeding depression (see Shea, 1987 for similar conclusions based on isozyme estimates of outcrossing in relation to tree size for subalpine fir and Engelmann spruce). What is important is the inter-play between the intrinsic variability available in the parent trees, and the way it is structured by contingencies of tree size and location. An useful adjunct to this study would be a molecular study of mating system. This would allow an independent test that the results we get in growth data from different canopy thirds are indeed a function of variation in the mating system. The purpose of this chapter has been to graphically explore the raw data and search for clues as to pattern and their possible sources. Based on these clues, the majority of the variation has been located as occurring with-in sites, specifically among progeny from different sections of the same tree. I have attempted some plausible hypotheses concerning some of the factors involved in this complex pattern. The detailed investigation of these hypotheses is beyond the scope of this thesis and will be pursued at a later date. The primary purpose of this chapter is to serve as a counter-weight to the developmental focus of subsequent chapters, by noting the complexity of variation in a natural stand and its dependance upon the interactions of multiple causes. 35 References Antos, J.A. and D.B. Zobel. 1986. Habitat relationships of Chamaecyparis nootkatensis in southern Washington, Oregon and California. Can. J. Bot. 64:1898-1909. Ayres, M.P. and D.L. Thomas. 1990. Alternative formulations of the mixed-model ANOVA applied to quantitative genetics. Evoln. 44:221-226 Banerjee, S. and Maze, J. 1989. Variation in growth with-in and among families of Douglas-fir through a single season. Can. J. Bot. 67: (in press). Brunei, D. and F. Rudolphe. 1985. Genetic neighbourhood in a population of Picea abies L. Theor. Appl. Genet. 71:101-110 Chambers, J.M., W.S. Cleveland, B. Kleiner, and P.A. Tukey. 1983. Graphical Methods for Data Analysis. Monterey, California: Wads worth. Clegg, M.T. Measuring plant mating systems. Bioscience 30:814-818 El-Kassaby, Y.A., A.M.K. Fashler and O. Sziklai. 1987. Effect of family size and number on the accuracy and precision of the estimates of genetic parameters in the IUFRO Douglas-fir provenance-progeny trial. For. Ecol. and Man. 18:35-48. El-Kassaby, Y.A., and R.H. Davidson. 1990. Impact of pollination environment manipulation on the apparent outcrossing rate in a Douglas-fir seed orchard. Submitted to : Heredity Fowler, D.P. 1965a. Effects of inbreeding in red Pine, Pinus resinosus Ait. II. Pollination studies. Silv. Genet. 14:12-23 Fowler, D.P. 1965b. Effects of inbreeding in red pine, Pinus resinosus Ait III. Factors affecting natural selfing. Silv. Genet. 14:37-46 Griffin, A.R. and D. Lindgren. 1985. Effect of inbreeding on production of filled seed in Pinus radiata — experimental results and a model of gene action. T.A.G. 71:334-343 Harry, D.E. 1987. Shoot elongation and growth plasticity in incense-cedar. Can. J. For. Res. 17:484-489 Khalil, M.A.K. 1987. Genetic variation in red spruce. {Picea rubens Sarg.). Silv. Genet. 36:164-171 Magnussen, S. and G.W. Yeatman. 1987a. Early testing of jack pine. I. Phenotypic response to spacing. Can. J. For. Res. 17:453-459 3 6 Magnussen, S. and G.W. Yeatman. 1987b. Early testing of jack pine. II. Variance and repeatability of stem and branch characters. Can. J. For. Res. 17:460-465. Maze, J., S. Banerjee, and Y.A. El-Kassaby. 1989. Variation in growth rate within and among full-sib families of Douglas fir. (Pseudotsuga meziesii [Mirb.] Franco). Can J. Bot. 66: (in press). Maze, J. and S. Banerjee. 1989. A comparison of variation between Pseudotsuga menziesii seedlings from genetically defined and undefined sources. McGill, R., J.W. Tukey and W.A. Larsen. 1978. Variations of box plots. The American Statistician. 32:12-16 Omi, S.K., and W.T. Adams. 1986. Variation is seed set and proportions of outcrossed progeny with clones, crown position, and top pruning in a Douglas-fir seed orchard. Can. J. For. Res. 16:502-507 Owens, J.N. and M. Molder. 1984. The reproductive cycles of western redcedar and yellow-cedar. Victoria, B.C. information Services Branch, Ministry of Forests, Province of British Columbia. Ritland, K. and Y.A. El-Kassaby. 1985. The nature of inbreeding in a seed orchard as shown by an efficient multi-locus model. Theor. Appl. Genet. 71:375-384 Scagel, R.K., Y.A. El-Kassaby and J. Maze. 1987. Multivariate variation within and between open-pollinated families of Douglas-fir (Pseudotsuga menziesii). Can. J. Bot. 65:1614-1621. Shaw, D.V. and R.W. Allard. 1987. Estimation of outcrossing rates in Douglas-fir using isozyme markers. Theor. Appl. Genet. 62:124-136 Shea, K.L. 1987. effects of population structure and cone production on outcrossing rates in engelman spruce and subalpine fir. Evolution. 41:124-136. Sniezko, R.A. and B.J. Zobel. i988. Seedling height and diameter variation of various degrees of inbred and outcross progenies of Loblolly pine. Silv. Genet. 37:50-60. c Strauss, S.H. 1987. Heterozygosity and developmental stability under inbreeding and crossbreeding in Pinus attenuata . Evolution 41:331-339 Wells, O.O. and J.R. Toliver. 1987. Geographic variation in sycamore (Platanus occidentalis L.) Silv. Genet. 36:154-159 Woods, J.H. and J.C. Heaman. 1989. Effect of different inbreeding levels on filled seed production in Douglas-fir. Can. J. For. Res. 19:54-59 Figure 1.1: Surnmary of sampling scheme. Tree 1-11 = midslope, 1 9 = swamp. Variation between Site (12) Tree (1-11.1-9) 3rd (1-3) Swamp (2) 3rd i 2.V 3 mod 38 Figure 1.2: Variation between sites represented by notched box-plots. The numbers in the upper right hand corner represent the percent of variation apportioned among groups as a ratio of sums of squares (LT 1% = "less than 1 percent"). Abbreviations: S = site (1 = midslope, 2 = swamp), H = height, T = time. T1 i o -10 H 5 -0 s T2 15 -H 10 5 -0 39 Figure 1.3: Variation among seedling families within each site as represented by notched box-plots. Abbreviations: Mi = midslope site, Sw = swamp site, Tr = Parent Tree of O.P. families. 40 Figure 1.4: Variation in seedlings among crown positions for selected trees from the midslope site as represented by notched box-plots. Abbreviations: Tr= Parent tree number seedlings originate from, 3rd = canopy third within parent tree seedlings originate from (1= top third, 2 = middle third, 3 = bottom third). T1 15 10 H T1 10 H Mi JJI3_ r / c S ? ^  T1 15 H 2 3 3rd M i Tr7 LT 1%! 8 ¥ K 2 3 3rd M i Tr11 T2 M i j t 2 . T2 M i Tr7 T2 M i Tr11 27J 3rd 3rd 41 Figure 1.5: Variation among crown positions for selected trees from the swamp site. Abbreviations as per figure 1.4. T2 1 8 1 2 H 12 1 5 1 0 H LIT/. - -1 2 3rd 1 1% • ^ v -* 1 L 3rd 42 Figure 1.6: Diagram plotting out positions of trees relative to an "average canopy". The lower dotted line is the average height to the crown (1st live branch) and the upper dotted line is the average total height of the trees. Tr = tree number. 1 2 3 4 5 6 7 8 9 Figure 1.7: Plot of germination capacity (G) against canopy position (3rd) for selected trees from the midslope (Mi) and swamp (Sw)sites. M i 44 THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR Chapter 2: Variation in Mainstem Growth and Needle Initiation Among and Within Open-pollinated Families of Yellow Cedar Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C., V6T 2B1 45 Abstract This chapter describes patterns of change as related to time in seedlings of yellow-cedar (Chamaecyparis nootkatensis) based on two data sets estimating developmental) growth of the mainstem and (2) initiation of needles in the mainstem. Seedlings from open-pollinated families were measured repeatedly over an eight month period, noting number of needle whorls initiated and height growth of the mainstem. Both height and needle whorl initiation data indicated that most of the variation resides within open-pollinated families. For both data sets the total variation increased with passage of time, primarily as a result of increasing variation within families. Variation among families also increased with passage of time, but to a lesser extent. Variation apportioned between the two sites was minimal and either increased slightly (whorl data) or decreased (height data). On a per-site basis, time trends of among family variation of whorl initiation data were similar for both sites, but differently structured among sites for height data. The characteristic increase in variation among individuals through time is interpreted in terms of diverging developmental trajectories, and is seen as a general consequence of ontogeny, as genetic and developmental differences between individual plants become manifest. 46 Introduction Hunt, (1981) has characterized plant development as concerning those "irreversible changes with time, mainly in size (however measured), often in form, and occasionally in number". While that description is seemingly simple, the actual underlying mechanisms are more complex. For example, many studies on plant growth deal with height growth, growth in length of the mainstem of a plant, an easily obtainable estimate of development. However the variation in such growth, that expressed in the leader in conifers, reflects variation in the plane and frequency of cell division and cell elongation in the different regions of the apex, such as, in conifers, the central mother cell, apical initial, peripheral zones, rib-meristem and their derivatives. In assessing growth of an organ the specific cell divisions and the elongation of that organ related to either cell elongation or subsequent divisions are confounded (Maze et al., 1989). An alternative assessment of development is to assess the initiation of a specific organ. For example, the growth of the leader in yellow cedar occurs concomitant with the initiation of needle primordia and then subsequent expansion of the internodes among needles. To some extent, the counting of initiation events is more focused on an different aspect of development than growth measures, the initiation of an organ being the product of a specific series of cell divisions in different planes, over a short time scale, relative to the subsequent elongation (or divisions) resulting in the growth of an organ. The question follows: are the two estimates of development congruent in generalities? In a series of studies seeking general patterns of change in ovule development (Maze et al., 1986, 1987, Scagel et al., 1985) two major 47 patterns emerged: ovule development is characterized by (a) increasing variation with time and .(b) intercorrelations changing between descriptors through time. A series of studies of leader and mainstem growth in Douglas-fir (Banerjee and Maze, 1988, Maze et al., 1989 and Maze and Banerjee, 1989) sought to cross-validate the ovule results to other plant structures, and at a different level of organization: genetically defined families. While the details differed, the generalizations based on the earlier studies of ovule development persisted: i.e. increasing variation and changing intercorrelations with time. In this study, these results are extended by examining two estimates of development: needle initiation and mainstem growth and their time dependent changes at the levels of open-pollinated families (O.P. families) and contiguous but ecologically different sites. Thus, the development of individuals is examined in the context of higher levels of organization (O.P. families, different sites) to see how development contributes to patterns observed at these higher levels. The specific questions I wish to address are the following: (1) To what extent can variation in plant size and needle-initiation number be apportioned to different sites, different parent trees within a site and differences among progeny from a single parent tree? (2) How does the apportionment of variation change with passage of time? What I wish to address is the degree to which individual plants become different from others with the same maternal parent, relative to those originating from different maternal parents and relative to plants from different sites. With time, do site, or parent tree effects become increasingly manifest; essentially, at what level does population differentiation occur? 48 The level at which among individual differentiation occurs ( within or between populations) has both theoretical and practical relevance. If differentiation can be attributed to the fact that different populations occupy different habitats, such information can be used as evidence that environmental selection determines the differentiation. However, if the bulk of the variation is from within population sources, this suggests that differentiation is either (a) a property intrinsic to individual organisms or (b) due to selection occurring for microhabitats, and would require a tight coupling between local environment and observed variation patterns. From the perspective of forestry applications, strong population differentiation implies that care must be taken in locating and delineating seed sources, whereas high within population variation indicates that one must assess the source of the within population differentiation and its genetic basis; an argument for individual tree selection. Materials and Methods Seed was collected from the top third of trees growing in two contiguous sites on Walker Mountain in the Sooke region of Vancouver Island (see Chapter One for details). One site was a swamp, the other was midslope. The seedlings were germinated and randomly sown into "4-15" styrofoam blocks with three replications of six seedlings for each open-pollinated family. There was an edge row which was not included in analyses. Five open-pollinated families originated from the swamp site while six open-pollinated families came from the midslope site. Needle initiation, as expressed in the number of whorls of needles present, was measured at thirteen different times. However, for the earlier times, the data were 49 highly skewed in distribution with extended tails below the median. Therefore analysis is presented only for times 7 to thirteen. Mainstem height was measured for five times. Dates for the measurement times are in Table 1. Patterns of variation between sites and between trees were summarized at each time using notched boxplots (Mcgill et al., 1978; Chambers et al., 1983) which graphically summarize variation within and between groups by plotting the medians, quartiles, outliers, and median confidence limits for a set of samples. They allow one visually to assess differences among groups in terms of median location, amount of variation about the median, and the skew of that variation. The amount of variation accounted for by sites and trees were also summarized using the analysis of variance (ANOVA). The ANOVA's were done in two stages, first one way ANOVA's were calculated for each source of variation, and then these ANOVAs were used to construct a nested ANOVA. Such an approach allows for a more detailed investigation of the variation attributed to each source than the use of a single ANOVA model, since it allows examination of the relative contribution of different groups at each level of the nested ANOVA. One way ANOVAs were done at each time on sites (model, Y = Sites + Trees(Site), and on trees within site on a per site basis (i.e. for each site, the model is Y= Trees + Individuals(Trees)) . Sums of squares (SS) for the one way ANOVA were used to construct a nested ANOVA by summing respectively the between and with-in tree terms for the one way ANOVAs done on each site. (Y = Sites + Trees(Site) + Individuals(Tree) ). The percentage of variation apportioned among and within open-pollinated families was expressed as ratios of SS (the eta^ statistic; O'Grady, 1982; Wilkinson 1978), the denominators being 50 respectively, the total variation of both sites, and the total variation in each individual site. Eta^, like the r^  statistic in regression analysis, is a measure of the variation explained (or apportioned to) a given term in a model. The additivity of SS allows for the construction of SS for a nested model from a series of one-way anovas. In doing these analyses I was concerned with time related changes in (i) total variation in the data set (ii) the amount of total variation accounted by sites, (iii) the amount of total variation accounted for by parent trees and (iv) the amount of variation among progeny from the same tree. Results Box plots are presented for three different times (to avoid an overabundance of figures) for each data type; for the whorl data, times 7, 10 and 13 and for the height data, times 1, 3 and 5. Figure One presents the variation between sites for the two data sets. Figure Two presents variation among trees for each of the two sites for the whorl data and Figure Three presents analogous information for the height data. In terms of both whorl and height data, there is an increase in amount of variation within each site. At time 7 the distribution of whorls is markedly skewed for both sites. However, the skew is below the median for the mid-slope site (site 1) and above the median for the swamp site (site 2). By time 10, the median value for whorls are similar for both sites, but by time 13 the medians are significantly different (just barely) with the midslope site material having the larger number of whorls. However, the variability about the median is increasing for the swamp site more rapidly than the 51. midslope, so that by time 13, though the medians are statistically different (i.e. the notches don't overlap) the quartiles of the swamp site overlap those of the midslope. The patterns of change between sites are somewhat simpler for the height data. The box-plots are moderately symmetric about the median at all times. Through time there is a switch in median rankings so that by time five the swamp site has the larger median value. Also, by time 5 the amount of variation about the median is approximately equal for both sites. In terms of variation patterns among the open-pollinated progeny of the parent trees sampled from the midslope site, there is little differentiation among progeny from different parent trees initially for either the whorl data (time 7) or for the height data (time 1). Again, the whorl data are initially quite skewed. For the height data, while there is an increase in variation through time, there is little change in the rankings of trees relative to each other, and all notches overlap, indicating no significant differences among medians. In the whorl data, however, there is a degree of differentiation among families, particularly between times 10 and 13. While there is no major switch in rankings, progeny from different trees become substantially different from each other; for example tree #1 relative to tree #2 and tree #3 relative to tree #4. Similar patterns appear in the whorl data on variation among open-pollinated families from the swamp site, with swamp tree #5 switching its relative position through time so that its median is greater than for tree #3 which had the largest value for median whorls at time 7. Again, with time, the data becomes less obviously skewed. In terms of height growth, there was considerably more among tree differentiation at time one in the swamp site than in the midslope site with progeny from swamp tree #5 showing 52 noticeably superior whorl production. While from times one to f i v e the relative median positions of progeny from trees #1 to #4 switch only slightly, progeny from tree #5 becomes substantially differentiated, and its interquartile range no longer overlaps that of the other trees. Progeny of Tree #5 also had superior whorl growth. The information in this series of box-plots is further quantified by a series of ANOVAs at each time (table 2-3). The details of these tables are summarized in figures 4-7 which allow us to examine the general pattern of time trends in differentiation at the following levels: among sites, among progeny from different trees within a site and among progeny from the same maternal parent. Figures 4-5 present absolute values of SS for the whorl and height data respectively, while Figures 6-7 present the variation among open pollinated families in each site as a ratio of the total variation in each site, comparison of the structure of variation in each of the two sites. For both whorl and height data the total SS, among open-pollinated families SS (nested model), and within open-pollinated families SS (nested model) increase with time. The among sites SS seems to be slightly rising with time (but fluctuates) for whorl data and steadily decreases for height data. In both cases, the variation apportioned between sites contributes little to the total variation. Most variation occurs among progeny from a given parent tree (i.e, within open-pollinated families; this is also the error term in the model), with a moderate amount of the variation further apportioned among progeny from different parent trees (15-34% for whorl data, 24-33% for height data; nested models). In terms of absolute SS, the among and within O.P. family terms show similar patterns through time for whorl data. For height data the patterns are less similar, and the total SS 53 primarily reflects the pattern of change with time in the within O.P. family term. On a per site basis, for the whorl data, the SS among trees increases for both sites. As a ratio of total variation (Fig. 6), there is, however, more fluctuation, but a general pattern of increase. Both sites show, for the whorl data, comparable levels of among tree differentiation. The height data, however, indicate a different structure of among tree differentiation for the two sites. For the midslope site, there is a decrease in the SS among-trees until time three, and a slight rise thereafter; essentially the same pattern is reflected when among tree variation for the midslope site is presented proportional to the total variation for that site. For the swamp site there is a steady increase in the absolute SS, but in terms of ratios of SS there is a fluctuating decrease. This implies that in the swamp site, while variation in height growth apportioned among open pollinated families is increasing, it increases at a slower rate than the variation apportioned within open pollinated families. What is striking is the amount of difference in the among-tree term for the two sites. In the midslope site a relatively minor component of variation in height is accounted for by the fact that progeny came from different maternal parents. However in the swamp site a significant proportion of the variability (from 40 to 50%) is accounted for by this term. Note, for a nested model, the absolute SS represents simply the sum of the among tree terms for ANOVAs done on a per site basis, and the proportional variation accounted for by trees in the nested model is the average of the proportional variation accounted for by trees for each site. Thus, where the magnitude of with-in population variability is heterogeneous among sites, a common occurrence in population studies, certain sites may contribute proportionally more to magnitude of variation accounted for by the among tree terms. 55 Discussion The differences in patterns of variation between two developmentally linked variables such as whorl initiation and height growth is intriguing and may provide some insight into the interplay between development and genetics. The initiation of an organ can be seen as a discrete set of events occurring over a relatively short time scale, whereas the subsequent elongation of that organ occurs over a longer time scale. In the case of yellow cedar, it was observed in the course of measurement, that some plants followed a pattern where a whorl of leaves was initiated, and the internode elongated for a period before the next whorl of leaves was initiated, whereas in other plants there occurred a series of initiation events, with little internode expansion between initiation events, then subsequent elongation, perhaps from intercalary growth. It was also observed that in a family row that seemed to vary in height, there was often actually very little variation in number of whorls initiated. Maze et. al. (1989), have detailed some of the complex histological events resulting in "simple" height growth. While the genetic basis of the morphology of an organ, such as needles, may be similarly complex, it is possible that the "timing" of initiation is under relatively simple control. Indeed, other timing events, such as phenology, seem to be under simpler genetic control than morphology. Since the growth of an organ occurs over a longer time scale, it is more likely to be affected by contingencies in the local environment and the physiological status of the plant. Thus, though initiation of a whorl and internode elongation are developmentally linked, it is possible that they 56 occur under a different set of internal and external influences and can show different, even inconsistent patterns of variation with time. One surprising result was the difference in the structuring of the within population variation for the two adjacent sites in terms of the height growth data. In the midslope site, most within site variation apportioned to within open pollinated families, whereas in the swamp site much of the variation was apportioned among open-pollinated families. This is also reflected in the absolute SS. The SS apportioned among trees is higher by a factor of 10 for the swamp site relative to the midslope site, while the SS within trees are about 1.5 times greater for the midslope site relative to the swamp site (Table 3a). That is, in the midslope site, knowing the maternal parent of a particular seedling gave little information concerning the state of another seedling from the same maternal parent (high within open pollinated family variation) while in the swamp site, knowing the maternal parent of a seedling gave relatively more information on the state of another seedling from the same parent. Furthermore, with time, knowledge of the maternal parent became less and less a guide to the nature of the progeny. This difference in within site structuring of height variation appears to be related to the canopy structure of the parent trees. Topographically, the midslope site was far more complex than the swamp site, and the canopy was more open. This allows for the possibility that the progeny from any given maternal parent on the midslope site reflect the effects of many more pollen parents than are available in the more closed canopy of the swamp site. However, this argument raises a second question: why, then, is the structuring of within site variation so similar for the whorl data across both sites? Possibly, the whorl data are less influenced by genetic contributions from the pollen parents. There are two ways this might come about. 57 Possibly whorl initiation is strongly influenced by maternal genetic effects and thus less subject to the vagaries of the pollen parents. Another possibility is that the genetic basis of whorl initiation is simpler than that of height growth (i.e. it might be polygenic, but controlled by relatively fewer genes; and there may be dominance). As the genetic basis becomes less and less additive polygenic, the liklihood of the maternal parent's influence increases. While, given the data and results available, I can only speculate on the actual mechanism of different structuring between sites, the important point is that they show very different patterning of within site variation. In a typical nested ANOVA model, this fine structure of within site differentiation is obscured, the within and between family terms in the nested model reflecting an average of the variation within and between open pollinated families in each site. Furthermore , the structuring of this variation seems to depend on an interplay between the topography of the population structure, and how it affects, via the mating system, the structure of genetic variation. This suggests that the quantitative gentics of natural populations of conifers cannot be considered in isolation of the spatial structure of the stand. The proportion of variation apportioned between families provides an estimates the heritable variation for a trait. Such an assessment of heritable variation is always dependent on the genetic structure of a specific population since the degree to which variation in a trait is heritable depends upon gene frequencies which may vary from population to population, and implies that estimates of heritable variation averaged across many populations may not prove to be a reliable indicator of the heritable variation in any single population. This problem is not unique to population studies of open-pollinated families from various sites, but also appears in the analysis of between family variation in full sib 5 8 studies, where each family can show a unique pattern of within family variation (Banerjee and Maze, 1988; Maze et al, 1989, Maze and Banerjee, 1989). It has been observed in many quantitative genetic studies of plants and animals where repeated measurements were taken over a period of time that (a) the amount of overall variation with time increases (often primarily due to increases in the within family variance term) and (b) this change in variation is accompanied by changing estimates of the amount of heritable variation at each time, often with a decrease in the heritable variation. These two results have been interpreted in a variety of ways in the literature. The overall increase in variation has often been cited as an "allometric effect" requiring appropriate scaling (often a logarithmic transform), or as a result of environmental variation becoming manifest. The changing heritabilities have also been suggested to reflect environmental interactions, and show the action of different groups of genes with time (see Harry, 1987; Nienstadt and Riemenschneider, 1985; Foster, 1986; Heuhn et al., 1987; Namkoong et al., 1972; Namkoong and Conkle, 1976;Silen, 1981, for the various ways changing heritabilities and juvenile mature correlations have been explained in the forest genetics literature). However the relevant patterns (a) increasing variation and (b) changes in the structure of variation (which would imply changing heritabilities) have also been observed in a series of multivariate studies of ovule development, in which the influence of environmental and quantitative genetic factors is minimal. Thus, if the general phenomena hold over a number of different levels, not all of which are governed by quantitative genetics, the explanation for the phenomena must be found outside quantitative genetics (which does not, however, imply that there is no genetic basis). This 59 observation leads me to suggest an alternate developmental interpretation for these phenomena, based on arguments initiated by Kaplan (1975) and further elaborated by Maze and Scagel (1983) in the context of studies of comparative development. I suggest that the increasing variation among individuals reflects primarily the differential expression of genes in an individual, as opposed to changes in the quantitative genetic structure of a population which would depend on genetic differences among families. Recently our group (Banerjee et al., 1990), utilized the idea of developmental trajectories, in the context of a multivariate space, to explain the increasing variation and changing correlation structure that characterizes development. The argument presented was that if the motion of an individual plant was viewed as a point trajectory through multivariate space (i.e. where the axes are the measured variables), then increasing variation would be determined by the point trajectories of individuals diverging with time (see Chapter 4 for a more detailed elaboration of these ideas). Furthermore, if the point trajectories diverged in a non-linear manner, the shape of the point swarm would change, leading to changes in correlation structure. A simpler version of this view, to deal with the univariate data of this study would be the following. Imagine a bivariate space, with time as the X axis, and our measured variable as the Y axis. The growth curve of each individual can be plotted on this graph. Where the growth curves diverge, variation increases with time. In a study where there is structuring above the level of the individual, a reasonable question to ask is, at what level is the divergence greatest? If it is greatest among individuals within a family it could be caused by (a) genetic differences among individuals through genetic recombination and (b) developmental differences among individuals reflecting differential gene expression leading 60 to a different ontogenetic history for each individual. If divergence is greatest between families, it could be attributed to the expression of quantitative differences between families. The results presented here argue for the divergence occurring at the level of individual plant developmental trajectories. While this difference between individuals, certainly partly reflect genetic differences between sibs within an O.P. family, it also may have a purely developmental component. Scagel et. al. (1985) argued, "an increase in variation is to be expected as individual ovules, once they have been initiated, will have independent ontogenies. Nonlethal developmental events that appear in one ovule will not necessarily appear in others". That is, even given similar or identical genotypes, the expression will lead to divergent developmental trajectories. A simple test of the argument that the increasing variation reflects differential gene expression leading to different developmental trajectories would be to examine patterns of variation for developing systems where all individuals are genetically very similar, either because they are clones (e.g. Poa bulbosa) or the result of heavy inbreeding (e.g. Arabidopsis thaliana). The implication is that the within family component of any genetic study encompasses a degree of developmental variation which is constantly changing with time (even in a constant environment), and affects estimates of heritable variation, by constantly altering, usually decreasing, the covariances between sibs. As long as variation within a family increases faster than variation between families (i.e. mean trajectories), estimates of heritable variation will decrease. At this point, I would like to comment on the relationship between increasing variation with time and growth rate from the perspective of developmental trajectories. Maze et al., (1989), in a study of growth rate in lateral branches in full-sib families of Douglas fir (Pseudotsuga menziesii) 61 found that most of the variation in growth rate apportioned within full sib families, specifically, the majority of variation apportioned among individuals (there were repeated measurements for each individual, so this was not the error term). As long as the developmental trajectories of all individuals in a population are roughly parallel, there will be relatively constant variation through time. However, for the trajectories to change so that variation increases with time, growth rates must change. Thus, variation in growth rates is required for developmental trajectories to change. This leads to a rather simple prediction. The rate of change in variation with time for a sample of individuals (slope of a plot of variance against time) will be positively correlated with the variance in growth rates of those plants. This is simply a syllogism, both analyses measure the same thing in different ways. Thus, the slope of within family variation measures the degree to which individuals within a family have different trajectories (or for a single variable, different growth rates). While an analysis of growth rates relies on a particular model of growth (usually logarithmic), Maze et. al. (1989) noted that not all individuals will correspond to a single curve shape. Calculating slope of the variance estimates the same quantity, variation among growth rates, without positing strong assumptions regarding the shape of individual growth curves. In this context, we can also ask a question that parallels one in the isozyme literature. What is the relationship between growth rate, and the rate of increase in variation?. From the point of view of development, there should be a positive relationship between growth rate and rate of variation increase since a more rapid increase in variation allows for more developmental events per unit time. In this view, it is not heterozygosity per se, but the putative ability of heterozygosity to result in faster growth rates that leads to increased 62 variation. These ideas are more appropriately studied on plants in which the genetics of growth are better understood than in conifers. C o n c l u s i o n The main results of this study were that while two estimates of development may have differences in details, they both exhibit the same general pattern with time of increasing variation, most of which apportions within open-pollinated families. Such a pattern of within group variation, seems to be carried down to the level of full sib families, and seemingly reflects the heterozygosity of the parent trees. The amount of within family variation, if it extends to older material, suggests that breeding can change population structure only at rather high selection intensities, and thus at the risk of greatly reducing genetic variation. Secondly, the fact that phenotypic variation continuously increases through time, implies that time to time correlations of family means which are often used to set up regimes of early selection will drastically underestimate the actual variability in the data, and hence the variability of response to selection. Current models of the relationship between phenotypic and genetic variation (Falconer, 1981) have no explicit role for the effect of development in the expression of any inherent genetic variation. Rather, "accidents" or "errors" of development are seen to be a source of "intangible" non-genetic variation, that can not be eliminated by experimental design (Falconer, 1981:125). Since developmental variation is intrinsic to organisms, it can not really be eliminated; it needs to be explicitly accounted for. 63 References Atchley, W.R. 1987. Developmental quantitative genetics and the evolution of ontogenies. Evolution 41:316-330 Banerjee, S. and J. Maze. 1988. Variation in growth rate within and among families of Douglas-fir through a single season. Can. J. Bot. 66:2452-2458 Banerjee, S.,J. Maze and P.R. Sibbald. 1990. Quantifying the Dynamics of Biological order and organization. J.Theor. Biol. 143:91-112 Chambers, J.M., W.S. Cleveland, B. Kleiner and P.A. Tukey. 1983. Graphical Methods for Data Analysis. Duxbury Press, Boston. Creighton, K.D. and R.E. Strauss. 1986. Comparative patterns of growth and development in critocene rodents and the evolution of ontogeny. Evolution 40:94-106. Foster, G.S. 1986. Trends in genetic parameters with stand development and their influence on earlv selection for volume growth in loblolly pine. For. Sci. 32:944-959 Harry, D.E. 1987. Shoot elongation and growth plasticity in incense-cedar. Can. J. For. Res. 17:484-489. Huehn, M., J. Kleinschmidt, and J. Svolba. 1987. Some experimental results concerning age dependency of different components of variance in testing Norway spruce {Picea abies (L.) Karst.) clones. Silv. Genet. 36:68-71 Hunt, R. 1981. Plant growth curves.The functional approach to plant growth analysis. John Wiley and Sons. Edward Kaplan,D.R. 1975. Comparative developmental evaluation of the morphology of unifacial leaves in the monocotyledons. Bot. Jahrb. Syst. Pflanzengesch. Pflanzengeogr. 95:1-105. Maze, J., Banerjee, S. and El-Kassaby, Y.A. 1989. Variation in growth rate within and among full-sib families of Douglas-fir {Psuedostsuga menziesii). Can. J. Bot. 67:140-145 Maze, J. and S. Banerjee. 1989. A comparison of Pseudotsuga menziesii seedlings from genetically defined and undefined sources. Can. J. Bot. 67: 945-947. Maze, J., K.A. Robson, S. Banerjee, L.R. Bohm and R.K. Scagel. 1990. Quantitative studies in early ovule and fruit development: developmental constraints in Balsamorhiza sagittata and B. hookerii. 64 Maze, J., and R.K. Scagel. 1983. A different view of plant morphology and the evolution of form. Syst. Bot. 8:469-472. Maze, J., R.K., Scagel, L.R. Bohm and N.L. Vogt. 1986. Quantitative studies of early ovule development. II. Intra- and inter-individual variation in Stipa lemmonii. Can. J. Bot. 64:510-515 Maze, J., R.K., Scagel and L.R. Bohm. 1987. Quantitative Studies in ovule development. III. An estimate of shape changes in Phyllostachys aurea. Can. J. Bot. 65:1531-1538. McGill, R., J.W. Tukey, and W.A. Larson. 1978. Variations of boxplots. Am. Stat. 32:12-16. Namkoong, G. and M.T. Conkle. 1976. Time trends in genetic control of height growth in Pinus ponderosa. For. Sci. 22:1-12. Namkoong, G., R.A. Usanis, and R.R. Silen. Age-related variation in genetic control of height growth in Douglas-fir. Theor. Appl. Genet. 42:151-159. Niendstaedt, N. and D.E. Riemenshneider. 1985. Changes in heritability estimates with age and site in white spruce Picea glauca (Moench) Voss. Silv. Genet. 34:34-41. O'Grady, K.E. Measurements of explained variance; cautions and limitations. Psychol. Bull. 92:766-777 Owens, J.N. and M. Molder. 1984. The reproductive cycles of western redcedar and yellow-cedar. Information Services Branch, Ministry of Forests, Province of British Columbia. Scagel, R.K., J. Maze, L.R. Bohm, and N.L. Vogt. Quantitative studies in early ovule development. I. Intraindividual variation in Nothofagus aurea. Can. J. Bot. 63:1769-1778. Silen, R.N. 1981. Nitrogen, corn and forest genetics. U.S. For. Serv. Gen. Tech. Rep. PNW-137. Wilkinson, G.N. 1978. ANOVA for non-orthogonal data. In Computer science and statistics: symposium on the interface. Proceedings of the 10th annual symposium held at the National Bureau of Standards, Gaithersburg, MD, April 14-15, 1977. Edited by, D. Hogbum, and B.W. Fife. NBS Spec. Publ. (U.S.) No. 503, pp. 58-64. Williams. C.G. 1987. The influence of shoot ontogeny on juvenile-mature correlations in Loblolly pine. Forest Science 23:411-422. Table 2.1: Measurement times for (a) whorl initiation and (b) height data. (a) Whorl I n i t i a t i o n Time Date (1987 -1988) 1 Oct 13 2 Oct 24 3 Oct 27 4 Oct 30 5 Nov 4 6 Nov 8 7 Nov 11 8 Nov 18 9 Nov 23 10 Nov 30 11 Dec 9 12 Dec 27 13 Jan 12 (b) H e i g h t Time 1 J an 31 2 Feb 12 3 Feb 27 4 Mar 25 5 Apr 14 66 Table 2.2: Whorl initiation data. (a) Sums of Squares apportioned to various putative sources of variation. "Trees" refers to the parent trees from which open-pollinated families were derived. TIMES 7 8 9 10 11 12 13 T o t a l 96. .2 175. .0 183. .6 239. ,5 284. .6 578. .6 863. , 4 Between S i t e s 2. .6 0. .1 2. .8 5. .1 3. . 2 8. .4 15. .5 Within S i t e s 93. .6 174. .9 180. ,8 234. ,3 281. .3 570. . 3 848 . .0 Among Trees 14. .1 36. .9 47. .5 56. .3 72. .6 162, .5 295. .2 Swamp 8. .6 19. .9 17. .2 24. .1 29. .3 62. .6 124 . . 4 Midslope 5. .5 17, .0 30. . 3 32. .2 43. .3 99, .9 170, .9 Within Trees 79 . .4 138 .0 133 . .3 178. .0 208 . .7 407, .7 552, .7 Swamp 30. .7 76. .3 49 , .0 56, .9 77. .3 127, .6 200, .9 Midslope 48. .7 61. .7 84, .3 121, . 2 131. . 4 280, .1 351, .8 0 67 (b) Percentage of total variation, based on ratios of sums of squares (eta2) from table la for which various sources account. TIMES 7 8 9 10 11 12 13 Between S i t e s 2. .8 0. . 0 1. . 5 2. 2 1 .1 1. 4 1. .8 W i t h i n S i t e s 97. .2 100. .0 98. .5 97. 8 98 .9 98. 6 98. .2 Among Trees 14. .7 21. .1 25. .9 23. 5 25 .5 28 . 1 34. , 2 Swamp 8. .9 11. . 4 9 . 4 10. 1 10 .3 10. 8 14. .4 Mi d s l o p e 5. .7 9 . . 7 16. .5 13. 5 15 .2 17. 3 19 . . 8 W i t h i n Trees 82. . 5 78. .9 74 . .1 74. 4 73 .4 70. 5 64. .0 Swamp 31. .9 43. .6 26. . 7 23. 8 27 . 2 22. 1 23. . 3 Mi d s l o p e 50. .5 35. .2 45. .9 50. 6 46 .2 48. 4 40 . 8 6 8 (c) Variation within each site represented as sums of squares (SS) and the percentage of variation within each site apportioned among and within open o pollinated families based on ratios of sums of squares (eta-6). TIMES 7 8 9 10 11 12 13 SS Swamp 39. 3 96. 2 66 .2 80.9 106.6 190.23 325. 3 Among trees 21. 8 20. 7 26 . 0 29.7 27.5 32.9 38. 2 Within Trees 78. 2 79 . 3 74 .0 70.3 72.5 67.1 61. 8 SS Midslope 54. 2 78. 7 114 . 6 153.4 174.7 380.1 522. 7 Among Trees 10. 2 21. 6 26 . 4 21.0 24 . 8 26 . 3 32. 7 Within Trees 89 . 8 78. 4 73 .6 79.0 75.2 73.7 67 . 3 69 Table 2.3: Height growth data. (a) Sums of squares apportioned to various putative sources of variation. "Trees" refers to parent trees from which open-pollinated families were derived. TIMES 1 2 3 4 5 T o t a l 5 1 6 . . 2 7 0 8 . . 1 8 4 7 . . 7 9 5 5 . . 6 1 1 2 5 . . 0 Between s i t e s 2 2 . . 7 1 6 . , 6 8 . , 5 5 . ,1 0 . . 4 Within S i t e s 4 9 3 . . 5 6 9 1 . . 5 8 3 9 . . 2 9 5 0 . . 6 1 1 2 4 . . 6 Among Trees 1 7 1 . .6 1 6 9 . . 7 2 1 7 . .8 2 2 5 . .5 2 8 7 . .3 Swamp 1 3 9 . . 0 1 5 2 . . 2 2 0 6 . . 1 2 1 1 . . 7 2 7 3 . . 3 Midslope 3 2 . . 7 1 7 . c . ~ l 1 1 . .7 1 3 . .8 1 4 . .0 Within Trees 3 2 1 . . 9 5 2 1 . . 8 6 2 1 . .4 7 2 5 . , 0 8 3 7 , . 3 Swamp 1 3 4 . . 7 1 9 2 . . 6 2 2 5 . . 3 3 0 2 . ,1 3 2 9 . . 1 Midslope 1 8 7 . . 1 3 2 9 . . 2 3 9 6 . . 1 4 2 3 . .0 5 0 8 . .2 70 (b) Percentage of total variation, based on ratios of sums of squares (eta2) from table 2a for which various sources account. TIMES 1 2 3 4 5 Between S i t e s 4 . , 4 2. . 3 1. . 0 0. . 5 0 . ,0 Within S i t e s 95. . 6 97. . 7 99 . . 0 99 . . 5 100 . 0 Among Trees 33. .2 24 . 0 25. . 7 23. . 6 25. . 5 Swamp 26. .9 21. . 5 24. .3 22. .2 24 . 3 Midslope 6 . 3 2. . 5 1. . 4 1. . 4 1. .3 Within Trees 62. . 4 73. . 7 73. . 3 75. .9 74 . 4 Swamp 26 . , 1 27 . 2 26. , 6 31. .6 29 . . 3 Midslope 36 . 3 46 . 5 46 . 7 44 . 3 45. . 2 71 (c) Variation within each site represented as sums of squares (SS) and the percentage of variation within each site apportioned among and within open-pollinated families based on ratios of sums of squares (eta2). TIMES 1 2 3 4 5 SS Swamp 273.7 344 . 8 431.4 513. 8 602.4 Among Trees 50.8 44.1 47.8 41. 2 45.4 Within Trees 49.2 55.9 52.2 58. 8 54.6 SS Midslope 219 . 8 346 . 7 4 0 7.8 436. 8 522. 6 Among Trees 14.9 5.0 2.9 3. 2 2.7 Within Trees 85.1 95.0 97.1 96. 8 97.3 Figure 2.1: Variation between sites, for seedlings, represented by notched box-plots for times 7, 10 and 13 of whorl data (a-c) and times 1, 3 and 5 of height data (d-f). Whorl Data a ) 25 1 5 10 Height Data T1 24.0 18.5 I-13.0 J-b) 20 T3 24.0 18.5 13.0 C) 15 10 T5 Mi S w 73 Figure 2.2: Variation among seedlings from O.P. families for the midslope site represented by notched box plots for whorl data (a-c) and height data (d-f). 24.0 16.5 13.0 -Whorl Data a ) Height Data 1 2 3 4 6 6 d ) 1 2 3 4 5 6 b ) 24.0 18.5 13.0 7.5 2.0 1 1 1 1 1 1 T10 25 20 1 1 1 1 1 1 T3 * * $ £ ^  £ A 15 - i i , i - 10 * e ) 1 2 3 4 6 6 1 2 3 4 5 6 24.0 18.5 13.0 f) 1 2 3 4 5 6 1 2 3 4 6 6 74 Figure 2.3: Variation among seedlings from O.P. families for the swamp site represented by notched box-plots for whorl data (a-c) and height data (d-f). a ) 24.0 18.5 13.0 7.5 2.0 W h o r l D a t a I 1 17 i i i i i i * Height D a t a 1 2 3 4 5 24.0 18.5 b ) 13.0 h 75 r-2.0 -i 1 r T13 _ l i l _ 25 20 15 10 1 2 3 4 5 6 1 2 3 4 5 6 24.0 18.5 13.0 C) 75 r-2.0 1 1 1 T 1 0 1 1 * 1 1 1 1 1  1 2 3 4 5 0 1 2 3 4 5 6 75 Figure 2.4: The apportionment of variation in seedlings for whorl data (Wh) as SS. The vertical axis is the square root of SS from a nested ANOVA and the horizontal is time. To = total variation. Si = variation due to sites. Fa = variation due to open-pollinated families, Se = variation due to seedlings (the error term in this model). Time Time 77 Figure 2.6: Variation among open-pollinated families in whorl data for the midslope (Mi) and swamp sites, as the percentage of variation within the site accounted for (Eta^5). Q J :  Time 78 Figure 2.7: Variation among open-pollinated families in height data for the midslope and swamp sites as the percentage of variation within site accounted for (Eta )^. "i me 79 THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR Chapter 3: A Pilot Study on the Nature of Morphological Integration Within and Among Individuals Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C., V6T 2B1 80 Abstract This chapter is a pilot study investigating variation in morphological integration in needle (from seedlings) and scale (from mature trees) morphology of Yellow cedar. Summary statistics were used to describe intra-individual variation in terms of the amount of morphological integration, reflected by the strength of variable intercorrelations, and the structure of morphological integration, reflected by the specific pattern of correlations. Approximately 22% of the variation in the structure of morphological integration for needle data was apportioned among open pollinated families, while ca. 41% of the variation in the amount of morphological integration was apportioned within families. For the scale data, measures of integration were calculated for crown thirds within each of three trees, and each tree showed a unique pattern in terms of amount and structure of integration within the crown. Variation in the measured variables and a synthetic variable, the first axis of a principal component analysis, was summarized using nested ANOVA models. For needle data from seedlings the most of the variation apportioned among individual seedlings, while for the scale data from mature trees most of the variation was apportioned within trees. These results demonstrate that patterns of covariation within an individual can change in the course of ontogeny so that the production of variation during the course of development is reflected by unique patterns in the structure and amount of morphological variation within individuals. 81 Introduction The concept of morphological integration was introduced by Olson and Miller (1958) who hypothesized that the degree of correlation among parts of an organism represented either functional or developmental relationships among those parts. Recent studies of morphological integration continue the theme developed by Olson and Miller, but most have concentrated primarily on the statistical aspects of morphological integration, primarily the elaboration of measures of morphological integration for multivariate data sets ( Cheverud 1982, 1988, 1989; Cheverud et al., 1983,1989; Wagner, 1984; Zelditch 1987, 198 and tests of the statistical significance of these measures, either against a null hypothesis of no integration (Wagner, 1984), or a specific developmental model (Zelditch, 1987, 1988). Most recent studies of morphological integration have focused on variation among individuals, or in quantitative genetic studies, on variation among family centroids. Yet development is not a property of populations, but of individuals. Hence, in this study, it is the morphological integration of individuals that is examined, as expressed in the nature of correlations among needles (in seedlings) and scales (in mature plants) within individual plants of Yellow cedar. By "nature of correlations", what is meant is summaries of correlation matrices which describe the relationship among needle or scale variables sampled from a localized area within each plant. In this context morphological integration is the result of developmental co-variability, the covariation within an individual that emerges as development unfolds. From the perspective of events occurring in an individual, it is possible to ask a series of questions about the nature of 82 morphological integration that can not be addressed in studies which are focused on correlations based on inter-individual (family means, populations) differences. / The first question is whether morphological integration is constant among and within individuals. This question can be restated in terms of apportioning morphological integration among individuals, and either spatially or temporally separated parts of individuals. The approach used is analogous to apportioning variation in individual attributes ( e.g. needle length) to various hierarchical levels with a nested ANOVA (see Chapter 2). The second question is whether morphological integration itself has a component of heritable variation. In this case what is being examined is not variation for a single metric trait, but variation for a measure summarizing an aspect of the relationship among traits within an individual. In addressing these questions I am concerned with two aspects of morphological integration. The first concerns the amount of morphological integration. This can be interpreted as an assessment of the strength of variable inter-relationships and is revealed through a summary measure of a correlation matrix. The second aspect of integration is its structure, the specific pattern of correlations in a matrix (i.e. the variation among individual correlation coefficients). Another manifestation of this structure is the orientation of data points in the space of the measured variables (Banerjee et al., 1990). Thus structure may be estimated by reflecting the orientation of points in variable space. Both amount and structure of integration can be seen as referring to specific aspects of a correlation matrix. They can also be seen as related to the concept of "organization", since biological organization is recognized by coherent interactions among parts, which are assessed by correlation 83 coefficients. To some extent "integration" and "organization" seem to be used in the literature as synonyms, though organization seems to be most closely associated with the structure of correlations, rather than the overall amount or correlations; measures of "integration" usually rely on statistics that describe average values of correlation coefficients. The notion of "integration" and "organization" can be considered primitive terms, representing features of living organisms that are intuitively obvious to biologists, but difficult to define precisely (Nagel, 1961; Rosenberg, 1985). My concern here is the mathematical representation of a concept that is well entrenched in biology, but still unclearly understood. The notion, however precedes the mathematical measures. The amount of integration among parts, and the actual pattern of inter-variable relationships (structure or organization) are logically distinct. For example, assume that correlations between leaf length and width data for two plants are of similar magnitude but in one plant the relationship is positive, in the other it is negative. Thus, though the amount of integration is of similar magnitude, it is structured rather differently for the two plants. This argument can be elaborated upon in the context of data sets with more than two variables. The implication is that for any given amount of integration, there can be numerous different correlation structures that will achieve that degree of integration. Thus, in this chapter, I examine patterns of variation in estimates of the amount and structure of within individual integration. The complexity of analyses require large within individual sample sizes, thus what follows is a pilot study based upon a subsample of the data available. Its purpose is to determine what general trends emerge in patterns of within individual 84 integration, which can be investigated in more detail in a more comprehensive study. A final point should be noted. The perceived integration among parts occurs through the course of ontogeny. However, in this study, ontogeny was not directly assessed via repeated measurements in time as in Chapter 2; rather the patterns of variation of the mature products were assessed and examined to determine to what type of ontogenetic framework they were consistent. This is an indirect approach to development. Rather than examining a temporal sequence of developing parts, what is examined is a spatial sequence of the mature products of development. These points are further elaborated in the discussion. 85 Materials and Methods Measurements: The variables measured on mature plants and seedlings are displayed in fig l(a-b). While the mature foliar organs are scale-like in appearance, the juvenile foliar organs are awl-shaped needles; maturation seems to result from the juvenile needles being reduced and appressed. For mature plants, needles were sampled from adjacent branchlets along the longest branch within each third of the crown (top, middle, bottom). For each crown third, 5 sets of 4 to 6 adjacent scales were sampled from a branchlet. Thus, the sample size within each crown third is ca. 20 - 30 individual scales, from a fairly restricted portion of the plant. Variables were measured (Fig. #la) that represented linear distances assessing the length and width of scales. On the scale data, six variables were measured initially, but for the purposes of subsequent analysis, only three are used: "SCALE LENGTH", "SCALE WIDTH" and length from widest point on scale to scale tip ("HALF LENGTH"). This reduced data set was used since it corresponds to measurements taken on a single discrete organ, the scale, rather than encompassing measures relating to the relationships among scales. In this pilot study, the analysis is restricted to that representing the canopy thirds for three trees, two of which were also parent-trees for two of the six open-pollinated families studied for juvenile data. A pilot study such as this must rely first upon a restricted sampling to discern if pattern may be present, following which the details of that pattern may be worked out in a more extensive data set. 86 For seedlings, juvenile form needles were taken from the mainstem. Usually 15 successive needles were taken after counting up five whorls from the cotyledons. This procedure was followed so that all samples were from developmentally comparable regions of the plant. Six open-pollinated families, with 2 -4 (for 4 of 6 families)individuals/family are represented in the data set. The four variables measured on needles assess the "LENGTH", and width at three points on the juvenile needle; the middle of the needle ("WIDTH TWO") and half-way to the needle tip ("WIDTH ONE") and half-way to the needle bottom ("WIDTH THREE"; see Fig lb). These measures of external morphology are not as precise estimates of developmental events as anatomical features used in previous studies of intra-individual variation in conifers (Chen et al., 1986; Maze et al., 1986, Maze et al., 1990) but have the advantage that they are relatively easy to measure. The switch from juvenile to mature needle form would provide a fascinating case for an in depth developmental study at the anatomical level. All measures were made using the Contron Image Analyzer in conjunction with a dissecting microscope. Needles and scales were mounted on cardboard, the image was projected on a video screen, and measures were made on the enlarged video image. Thus, the measures were made on two dimensional images of the needles and scales. This approach, while efficient, has the potential of producing measurement error resulting from three dimensional images flattened into two dimensions. 87 Analysis: For each crown third of the mature trees, and for each seedling, the properties of the correlation matrix of descriptors of needle morphology were summarized by two measures. The first summary measure was the determinant of the correlation matrix, |R|. |R| is a scalar summary of the strength of variable intercorrelations (Green and Carroll, 1976). As correlations among variables increase in magnitude, |R| approaches 0. As correlations among variables decrease in magnitude, |R| approaches one. Thus, for a set of data where all variables were independent, |R| is 1; and where all variables are so strongly coupled that any variable can be estimated as a linear function of the other variables, JRJ = 0. |R| was represented as negative logarithms (-In |R|). In this form, low values mean lower integration, while higher values mean higher levels of integration, which corresponds well to our intuitive sense for a measure of integration. Hence, -ln|R| can be seen as assessing the amount of integration and can be related to mathematical measures of order (Banerjee et al., 1990). The second summary measure seeks to assess one aspect of the structure of correlations, the orientation of the major axis of variation based upon a principal components analysis of a correlation matrix (Pimental, 1979; Maze et al., 1987, 1990). A principal components analysis (PCA) is an eigenvalue decomposition of a correlation matrix, which seeks to summarize the correlation matrix as a series of uncorrelated vectors (eigenvalues) whose orientation through variable space is represented by a vector of direction coefficients (the eigenvector associated with an eigenvalue). The first (or major) axis from a PCA is that which best distinguishes among individual data points (in this case scales or needles). The orientation of this major axis 88 of variation for each individual seedling or crown third was represented by its angular deviation from (a) a vector of isometry ("THETAISO") and (b) the major axis of variation for a PC A of the correlation matrix based on the pooled data for all individuals ("THETAPCA"). In the first case ~ deviation from a vector of isometry — what is being assessed is the degree to which differences among individuals for the first PC A axis (that which maximally differentiates among individual data points) result from differences in shape, rather than size. Thus a swarm of data points well described by a vector of isometry would represent objects whose proportions are relatively constant with changes in size; whereas when a data set deviates from a vector of isometry, differences among individuals are both in size, and relative proportions among variables; i.e., "shape". Since a vector of isometry is one where all eigenvector values are of equal magnitude, what is assessed is essentially the heterogeneity of eigenvectors. Since these eigenvectors represent properties of the correlation matrix, their heterogeneity is related to the heterogeneity of the correlation coefficients in the matrix. The fact that two data sets have the same angular deviation from a vector of isometry does not imply that they have the same orientation. Thus, the second comparison of angles is relative to the pooled data set, and examines the degree to which the orientation of an individual in multivariate space is different from the major axis of variation examining orientation among all variables. Angles must always be based on some reference axis, and thus it was believed that two useful reference axes could be one with certain theoretical implications (the vector of isometry) and the other reflecting the structure of the data set as a whole (the first axis of a pooled PC A). Other reference axes could have been chosen. This method of characterizing vectors from PCA with other reference vectors has been used by those 89 employing multivariate analysis to describe the properties of data sets in terms of the concept of shape (Bryant and Meffert, 1988; Cheverud et al., 1983; Creighton and Strauss, 1986; Kohn and Atchley, 1988; Maze et. al., 1987, 1990). In the context of multivariate analysis, differences in the shape of objects are reflected as differences of orientation in the multivariate space in which those objects are projected. Hence angular comparisons of orientation may be seen to be estimated differences in organization for the groups of objects being compared. For the purposes of the results and discussion that follow, morphological integration is considered to have two aspects; the amount of integration, or its order, is assessed by -ln|R| while the structure of integration, or its organization, is reflected in the angular comparison (THETAISO and THETAPCA) which may be seen as assessing differences in organization. These two aspects of integration are logically distinct. The differences among seedlings for -ln|R|, THETAISO, and THETAPCA were summarized by one-way ANOVAs in which variation in these measures was apportioned among and within the four O.P. families. Nested ANOVAs were also used to summarize variation in the raw data, and the first axis of a PCA. For the seedling data the model was: Y= U + O.P. Family + Seedling(O.P. Family) + Needles(Seedling). For the Parent Tree data the model was: Y= U + Parent Tree + Third(Parent Tree) + Branchlet(Third) + Scales(Branchlet). 90 In both cases, "Y" represents the dependent variable subjected to analysis, either one of the measured variables or the major axis of the PCA on the pooled data set. The construction of these models and their presentation was along the lines discussed in Chapter 2, and the discussion therein can be referred to for further analytical details. The results of the ANOVAs were corroborated with notched box-plots, however these are not presented due to space considerations. 91 Results Nested ANOVA results for the juvenile data for PCA 1, and the four measured variables indicate that not a great deal of variation in the data is accounted for by open pollinated families (from 5 to 17 %; Table 3.2). For PCA 1, and needle length, ca. 54 and 62% of the variation is apportioned among individual seedlings. PCA 1 itself seems to be close to a vector of isometry with all positive eigenvector coefficients of similar magnitude, and this axis accounts for 57% of the total variation in the data set (Table la). For the first two width measurements from the needle tip, ca. 30 % of the variation is apportioned among seedlings, while for the third width measurement (nearest to the needle base) ca. 50% of the variation is apportioned among individual seedlings. Though individuals are often the "error" term in other genetic studies, it seems that this term can include a high degree of structured variation, indeed. For mature scale morphology, PCA 1 for the pooled data set (Table lb) had all positive eigenvector coefficients of similar magnitude and summarized 68.4 % of the total variation in the data set. The result of nested ANOVAs on the original variables and PCA1 for the scale data are presented in Table 3. Parent Trees accounted for from ca. 9 to 20% of the variation, crown thirds within parent trees accounted for ca. 10 to 30 % of the variation, branchlets within a crown section accounted for ca. 26 to 38% of the variation and variation among scales within a branchlet (which also includes measurement error) accounted for the remaining 27 to 43% of the variation. Overall, the model accounted for from 56 to 73% of the variation in the data, with most of that variation arising from within tree sources, primarily resulting from variation among replicate 92 branchlets within a crown section. It should be noted that such replicate branchlets were sampled from a region of a larger branch within inches of each other, and thus a large degree of the variation is apportioned among structures at a very local scale. Table 4 presents the results of the nested ANOVA for PCA 1 of the scale data in more detail so that the contribution of individual parent trees and crown thirds within a parent tree can be examined. The results for the measured variable (not presented) were similar to the results for PCA 1. Parent tree #1 contributed proportionally more to the variation among crown thirds than the other two parents. Furthermore, the proportional contribution of Parent trees to differentiation among crown thirds (Tree #1 > Tree #3 > Tree #2) is reflected again at the finer scale of differentiation among branchlets within a crown section where the upper third of the crown of parent tree #1 again accounts for proportionally more of the variation among branchlets than the upper third of the other two parent trees. There is no obvious trend in the amount of variation apportioned among branchlets in a given crown section, the apportionment of within individual variation seems to be unique to each tree examined. While the results presented suggest that there may be a relationship between within individual differentiation at a fine scale (e.g among branchlets) and within individual differentiation at broader scales (e.g among crown thirds), more conclusive results require a larger data set, and should include covariates such as the age of a branch that a branchlet is sampled from. Tables 5 and 6 present the results for the descriptors of the amount (-ln|R|) and structure (THETAISO, THETAPCA) of morphological 93 organization. Needle data appear in Table 5, while scale data appears in Table 6. For needle data variation in the summary statistics -ln|R|, THETAISO, and THETAPCA, was apportioned using one-way ANOVAs. The amount of variation apportioned among O.P. families for both the angle measures was ca. 22%, whereas the amount of variation apportioned for -ln|R| was ca. 41% (Table 7). However, the differences among families were not statistically significant, possibly related to the low sample size with-in families. In a related study on Douglas-fir (Maze et al., 1990), with much higher sample sizes, most of the variation in summary statistics (in this case -ln|R|) was apportioned within families, in general agreement with the present study. Interestingly, while both angle measures were highly correlated (r = .91; probably because PCA 1 was itself similar to a vector of isometry) the correlation between angles and determinants was very low (ca. r = 0.2), indicating that the statistics assess rather different features of the morphological data. In the scale data, the differences among all three summary measures of morphological integration were greatest for crown thirds in parent tree #1 and least in parent tree #3 (Table 6). The variation in these estimators of different aspects of integration seems to correspond to the ordering of variation in Table 4, which suggests that the production of variants is associated with changes in both the amount and structure of that variation. Such an hypothesis is perhaps best tested in the context of a data set where temporal variation in sequential structures can be assessed directly. 94 Discussion When making comparisons of variation among sequentially produced organs in parts of a plant that are spatially separated, what is being assessed is the end product of cell lineages that have diverged from each other at a specific point in time. A set of sequential needles represents a series of structures whose point of developmental divergence is recent, relative to needles gathered from widely separated parts of a plant. Thus, roughly, there is a correlation between the spatial separation of homologous organs in a plant and the degree to which they represent the products of cell lineages that have diverged from a common ancestral lineage. This relationship between the developmental origin of organs, and the strength of their intercorrelations seems to be supported by studies that compare sequential or juxtaposed organs with organs more distantly separated (Grafms, 1978; Morishima and Oka, 1968; Scagel and Maze, 1984) where it was found that spatially adjacent or developmentally sequential variables were more strongly correlated than those that were distant from each other. Similarly, in this study, the largest terms in the nested ANOVA model were those associated with variation among branchlets, where the branchlets sampled were gathered from a very restricted portion of the plant, and sequential scales were measured within each branchlet. To some extent the variation among branchlets may also reflect the indeterminate nature of growth in yellow cedar. Though the branchlets sampled were all closely spaced, they may have been initiated at different times and under different conditions. It would be interesting to compare these results to variation in needle morphology among branches of a conifer with a determinate growth 95 pattern (many of which show indeterminate growth in their first year, and can be induced to have indeterminate growth at later stages). Long lived perennials demonstrate a great degree of variation in their morphology (Blue and Jensen, 1988), and it is reasonable to enquire into what sources such variation originates. One hypothesis is that it is a reflection of genetic changes in somatic cell lineages in the course of development. Klekowski (1984, 1988; Klekowski and Kazarinova-Fukshansky 1984a,b; Klekowski et al., 1985, 1989) provides theoretical arguments and some empirical evidence (specifically, from ferns) that long lived plants may be fairly genetically heterogeneous as a result of somatic mutations over the course of their lifetimes, even when the probability of mutation/cell cycle is fairly low. A second hypothesis, briefly alluded to in Chapter 2 and more strongly developed in Chapter 4 considers epigenetic changes in the course of development, wherein cell lineages become increasingly independent of each other over the course of time. Thus, rather than somatic mutations, all that is required is a change in gene expression during the course of development. Two logical implications from such an idea follow: (1) that over time intra-individual variation increases and (2) part of the manifestation of increasing independence among cell lineages will be differences in the correlation properties of homologous organs. The results presented here are consistent with both of those deductions. In this study, it was demonstrated that there can be differences in the correlation properties within an individual as assessed by measures of integration and angular deviations for the mature morphology data. However, there was no clear trend to the pattern of how correlation properties change. This is not surprising, given that by sampling branches 96 from different regions of the canopy we can only say that cell lineages have diverged; draw no conclusions concerning the specific nature of the divergence, or even which is the older cell lineage, that at the top of the canopy or that from an mature branch at the bottom. The point is that the structure of covariation does change. The juvenile morphology provides an interesting contrast. Most of the variation in juvenile morphology apportions among individuals within families, while in the parent trees, most of the variation falls within individuals, and the greater part of this variation apportions among replicate branchlets within a crown third. Such results imply that the process of development from seedling to mature tree is coincident with the generation of a diversity of covariation patterns within trees that obscures the covariation patterns among trees.,This insight should be tempered by the fact that the intensity of within individual sampling was not identical for the two data sets. For an individual seedling only one sample of needles per individual was obtained and statistics were based upon that. For the parent trees 15 samples of scales were obtained per individual. To some extent the variation among individual seedlings and the variation among branchlets in the parent-trees represent comparisons among sets of sequentially produced organs, and it is at this level that much of the structured variation accrued in both cases. In terms of patterns of integration for the scale data, not only are individual parent trees different; but parts of individual trees are different. Each individual parent-tree seems to have a unique pattern of integrated covariation. The implication is that in the course of an individual's developmental history it is "individuated", and the byproduct of this is the unique patterns of variation perceived on the mature organism. 97 References Banerjee, S., P.R. Sibbald and J. Maze. 1990. Quantifying the dynamics of order and organization in biological systems. J. Theor. Biol. 143:91-112 Blue, M.P. and R.J. Jensen. 1988. Positional and seasonal variation in oak (Quercus; Fagaceae) leaf morphology. Amer. J. 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Kazarinova-Fukshansky. 1984b. Shoot apical meristems and mutation: selective loss of disadvantageous cell genotypes. Amer. J. Bot. 71:28-34 Klekowski, E.J., N. Kazarinova-Fukshansky and L. Fukshansky. 1989. Patterns of plant ontogeny that may influence genomic stasis. Amer. J. Bot. 76:185-195 Klekowski, E.J. N. Kazarinova-Fukshansky and H. Mohr. 1985. Shoot apical meristems and mutation: stratified meristems and angiosperm evolution. Amer. J. Bot. 72:1788-1800 Kohn, L.P. and W.R. Atchley. 1988. How similar age genetic correlation structures? Data from mice and rats. Evolution 42:467-481 Maze, J. S. Banerjee, and L.R. Bohm. 1990. A study of the correlation properties of needle attributes in full sib families of Douglas fir. Submitted: Amer. J. Bot. Maze, J., K.A. Robson, S. Banerjee, L.R. Bohm and R.K. Scagel. 1990. Quantitative studies in early ovule and fruit development: Developmental constraints in Balsamorhiza saggittata and B. hookeri. Submitted: Botanical Gazette. Maze, J., R.K. 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A morphological analysis of local variation in Stipa nelsonii and S. richardsonii (Graminea). Can. J. Bot. 62:763-770 Wagner, G.P. 1984. On the eigenvalue distribution of genetic and phenotypic dispersion matrices: Evidence for a nonrandom organization of quantitative character variation. Zelditch, M. 1987. Evaluating developmental models of integration in the laboratory rat using confirmatory factor analysis. Syst. Zool. 36:368-380 Zelditch, M. 1988. Ontogenetic variation in patterns of phenotypic integration in the laboratory rat. Evolution 42:28-41 Zelditch, M. and A.C. Carmichael. 1989. Ontogenetic variation in pattern of developmental and functional integration in skulls of Sigmodon fulviventer . Evolution 43:814-824 100 Table 3.1: PCA 1 for (a) needle and (b) scale data sets. (a) Needle Data PCA 1 E i g e n v a l u e 2.31 % v a r i a n c e 57.86 E i g e n v e c t o r : LENGTH 0.62 WIDTH1 0.78 WIDTH2 0.83 WIDTH3 0.78 (b) Scale Data PCA 1 E i g e n v a l u e 2.05 % v a r i a n c e 68.41 E i g e n v e c t o r : SCALE LENGTH 0.77 SCALE WIDTH 0.91 HALF LENGTH 0.79 Table 3.2: Nested ANOVA for measured variables and PCA 1 for needle data, (a) Sums of Squares (SS). (b) Percentage of variation as a ratio of sums of squares (Eta2). L=LENGTH, W1=WIDTH1, W2 = WIDTH2, W3 = WIDTH3 (a) PCA1 L Wl W2 W3 T o t a l 313 1381 15.0 16.0 14.9 Among F a m i l i e s 23 229 2.3 1.5 0.7 Among S e e d l i n g s 168 866 4.6 4.8 7.3 W i t h i n S e e d l i n g s 122 286 8.1 9.7 6.9 (b) Among F a m i l i e s 8 17 15 10 5 Among S e e d l i n g s 54 63 30 30 49 W i t h i n S e e d l i n g s 38 20 55 60 46 102 Table 3.3: Nested ANOVA for measured variables and PCA 1 for scale data, (a) Sums of squares (SS). (b) Percentage of variation as a ratio of sums of squares (Eta2). (a) PCA 1 LENGTH WIDTH HALFLENGTH T o t a l 198 30.4 4.27 5.38 Trees 21 4.3 0.85 0.47 T h i r d s 40 8.9 0.63 0.55 B r a n c h l e t s 73 9.0 1.11 2.05 R e s i d u a l 62 8.2 1.69 2.31 (b) ETA 2 (MODEL) 69 73 62 56 Trees 11 14 20 9 T h i r d s 20 29 15 10 B r a n c h l e t s 38 30 26 38 R e s i d u a l 31 27 39 43 Table 3.4: Nested ANOVA for PCA 1 of scale data (from Table 3) presented in more detail to include contributions of individual trees and crown thirds. "SS" = sums of squares. Eta 2 = ratio of SS Source/ SS Total * 100. SS ' ETA 2 T o t a l 197.485 Trees (T) 21.254 10.7 Thirds (Th) 40.185 20.3 T l 28.448 14.4 T2 1.449 0.7 T3 10.288 5.2 Branchlets 74.175 37.6 T l T hl 24.801 12.6 2 1.769 0.9 3 6.300 3.2 T2 T h l 1.605 0.8 2 4.761 2.4 3 6.526 3.3 T3 Thl 2.770 1.4 2 13.174 6.7 3 12.469 6.3 Residual 61.871 31.3 104 Table 3.5: -ln|R|, THETAISO and THETAPCA values for individual seedlings' needle data. F a m i l y S e e d l i n g - l n | R | THETAISO THETAPCA 1 1 1 2 1 3 1 4 0.817 0.807 0 .660 1.114 13.107 10.217 8.294 15.472 15.133 12.936 9.791 19.850 2 5 2 6 0.864 0.762 54.871 20.786 50.016 24.517 3 7 3 8 3 9 3 10 0.834 0. 624 1.743 1. 368 11.629 52.607 3.219 12.634 15.697 53.875 7.619 17.773 4 11 4 12 4 13 4 14 0.967 0.722 0.527 0.411 31.695 45.941 15.017 10.818 34.680 41.610 15.008 13.760 5 15 5 16 5 17 1.262 1.294 1.618 32.936 15.043 8.892 27.182 9 .213 8.021 6 18 6 19 6 20 6 21 1.527 1.559 0.420 1.499 11.474 62.255 41.289 11.973 14.307 65.224 43.728 9 .881 Mean: 1.019 23.341 24.277 S.D. 0.417 17.835 17.088 105 Table 3.6: -ln|R|, THETAISO and THETAPCA for individual crown thirds based on scale data. - l n | R | THETAISO THETAPCA Tree 1 T h i r d : 1 2.464 3.481 21.288 2 0.339 23.552 5.799 3 0.504 6.501 9.335 Tree 2 T h i r d : 1 0.652 11.093 6.585 2 0.224 7.630 15.239 3 0.497 17.700 1.838 Tree 3 T h i r d : 1 0.409 4.361 3.704 2 0.639 3.105 1.718 3 0.562 4.214 2.169 106 Table 3.7: ANOVA of variation in -ln|R|, THETAISO, and THETAPCA within and among open-pollinated families for needle data, (a) Sums of squares (SS). (b) Percent variation as a ratio of sums of squares (Eta2) (a) -ln|R| THETAISO THETAPCA T o t a l 3.485 6362 5839 Among F a m i l i e s 1.417 1365 1336 Re s i d u a l 2.068 4997 4503 (b) Among F a m i l i e s 40.7 21.5 22.9 R e s i d u a l 59. 3 78.5 77.1 107 Figure 3.1 Variables measured on (a) juvenile needles and (b) mature scales NEEDLES: 1 = LENGTH, .2 = WIDTH 1, 3 = WIDTH2, 4 = WIDTH3; SCALES: A = LENGTH, 2 = WIDTH, 3 = HALF LENGTH 108 THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR Chapter 4: Developmental Trajectories and the Ontogeny of Within Population Variation Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C., V6T 2B1 Abstract The development of the form of an individual organ through time can be described in terms of a point trajectory through multivariate space, a developmental trajectory. The nature of the developmental trajectory of an individual organ is ultimately a manifestation of cell division and elongation in various planes, resulting in the external form of the organ. When a population of developing individuals is described in terms of such trajectories, a number of different patterns can arise, and the nature of developmental trajectories of the individuals can be related to patterns of change in covariance structure of the population through time. In this context, three increasingly more complex models are graphically developed. The simplest possible model is one where, through time, there are only size increases in individuals, and all individuals grow at the same rate. In this case both covariance and correlation structures remain constant, and the only changes are in mean size. The second model is one in which individuals grow at different rates, but still maintain essentially the same shape (i.e. the relative growth rates of the parts are proportionally constant), in which case correlation structures remain constant, and changes are in mean size, and an increase in the variable covariances as a result of increased dispersion. Finally, when developmental trajectories diverge from each other in a nonlinear manner, changes can occur in both correlation and covariance structures, coincident with changes in size. A number of data sets are examined and are shown to correspond to the third model, changing means, covariances, and correlations associated with increasing variances. It is argued that the increasing variation is a property intrinsic to developing 110 systems, and results from the fact that at any point in the development of an individual, there are a number of different specific expressions of cell division and enlargement that could occur. Those events that occur in one organ/individual will not necessarily be those that occur in another. As these differences accrue over time, developmental trajectories diverge, leading to changes in the variation, covariation and correlation structure of the population of individuals under study.. I l l Introduction "Ask not what mathematics can do for biology, Ask what biology can do for mathematics" -Stanislaw Ulam Often it is the complexity and diversity of plant forms that generates an interest in comparative morphology. Yet it is an odd fact that the observed variety of plant forms originate through a relatively minor repertoire of events at the cellular level. The process of development in a plant is simpler than that in an animal, having fewer basic cell types, and lacking cell migration or, apparently, induction. Plant cells stay in place in relation to adjacent cells, and the development of a plant is a function of either division or elongation in a few planes. It is through the repeated iteration, and variable manifestation of these few basic events in terms of the timing and the various planes and positions of cell divisions and elongation events that plant development becomes manifest. How does such a limited repertoire lead to the great degree of variability observable even among individuals within the same species? The previous chapters have presented empirical results, demonstrating the complex patterns of variation found among individuals of yellow cedar, and the degree to which this complexity might have an environmental, genetic or developmental basis. We have examined the spatial complexity of the patterns of variation found, how their apportionment changes through time in seedling development, and the relationship between morphology in two generations. Throughout we have encountered the limitations of 112 classical quantitative genetic models in explaining these patterns satisfactorily. As noted in the prefatory chapter, my work was part of a larger research program that concerned general patterns in development and its manifestation at various hierarchical levels such as the individual, the population and the species. In this last chapter I would like to apply the results of the previous chapters, as well as a number of related studies by our research group, to the characterization of developing systems in plants as a starting point for elaborating, for the most part graphically, a model explanation as to how the patterns of increasing variation, and changing covariation may occur. In stating that I will attempt to provide a mechanistic explanation of development, I am, of course, participating in a degree of hubris. While any such explanation will be overly simplistic, and contain speculative elements, I consider my explanation to be no more simplistic or speculative than attempts of quantitative genetic models (and their recent extensions to natural populations) to explain differences among individuals in terms of a hypothetical quantity, the breeding value. My approach is to attempt to explain the patterns of increasing variation and changing covariation with development in terms of the notion of a developmental trajectory, or how individuals move through space with time. In this case, we are dealing with a "morphological space", which consists of the measured variables. At each time of measurement an individual can be located as a point in this space. When we make repeated measurements on the same individual over time the line connecting these points defines the developmental trajectory of that individual. We are concerned with the properties of a population of such trajectories. What I shall attempt is the following: develop three increasingly complex models of developmental 1 1 3 trajectories, describing the types of covariance structures, and the changes that these models would allow and then compare these models to patterns found in empirical studies. Secondly, I wish to provide an interpretation of the notion of "developmental trajectory" in terms more familiar to a comparative anatomist, and show how variation in development among organs and individuals can lead to divergent trajectories. Finally, I shall briefly consider how the models I use may be elaborated and formalized by using techniques from dynamical systems theory as applied to complex systems (see Stein, 1989; Nicolis and Prigogine, 1989, Campbell, 1989,) . For the most part, my presentation in this chapter is non-mathematical, a more mathematically oriented development of much of the same material appears in Banerjee et al. (1990) The notion of characterizing development in terms of "trajectories" has existed in the biological literature for some time, and is implicit in many comparative studies of development. In particular, Waddington (1957, 1977) developed the notion of a "chreod", or 'necessary path" in development, the idea being that development is canalized into certain paths or trajectories, and resists perturbations that might lead it to stray from those paths. Alberch et. al. (1979) attempted to formalize these ideas in terms of cross species comparisons. Specifically, they wished to model evolutionary events such as neotony, heterochrony and paedmorphism in terms of variations in developmental timing (specifically, differences in the initiation, cessation and rate of growth; see Kluge [1988] for a review of the development and application of these models), leading to different morphologies among related species. Their work was concerned primarily with differences between individual growth curves (usually representing a species mean) plotted 114 against time or age, or as trajectories in a two dimensional space, where one axis is "size" and the other "shape". In this two dimensional space, they attempt to give graphical definitions of the differences between neotony, progenesis, acceleration hypermorphis, paedmorphis and recapitulation for hypothetical ancestor descendant species pairs. Their approach has been expanded by a number of other workers, particularly in terms of producing characters reflecting development in a species that can be used for phylogenetic analyses (Alberch, 1985; Creighton and Strauss, 1986; Kluge, 1988 and Kluge and Strauss, 1985) and in attempts to apply quantitative genetics to ontogenetic data (Atchley, 1987,1989). While, most of the initial paper and later elaborations center around the discussion of such hypothetical changes in developmental trajectories with interspecific evolution, Alberch et aZ.(1979) briefly mention that "a population of maturing individuals can be represented by a 'cloud' of points moving through age-size-shape space"r but assume that covariance structures would change in such a space only under the influence of natural selection. While, their "age-size-shape" space is useful for providing heuristic examples, it is less applicable to empirical data, since both "size" and "shape" are abstractions. From a multivariate data set it is possible to extract a vector representing only pure size differences among individuals (a vector of isometry), but then, all subsequent vectors represent some aspect of shape. In many data sets, "shape" resides in no single variable, or ratio of variables but is a function of all variables. Thus, my models are stated in terms of a general variable space, rather than the more specific and less applicable age-size-shape space of Alberch et. al. Rather than concentrating on individual growth equations and their parameters, I concentrate on general patterns of developmental trajectories. 115 The simplest form of a developmental trajectory might be a growth curve. (Fig 1) in which one axis represents the variable measured, and the other, the times of measurement. Two hypothetical sets of curves, characterizing the growth of three individuals are presented in figure 1. In the first set of curves (Fig. la) the three individuals, have initial size differences, but grow essentially at the same rate. Thus, with time, while their size increases, there is no change in the dispersion of the group which they constitute. Such a situation could be represented as the result of iterating a series of growth equations (one for each individual in the population), whose only differences are in their intercepts. The second set of curves (fig. lb) again represents the development of three individuals. However, in this case, there are not only differences in initial conditions, but also differences in the growth rates. In such a situation there is a pattern of increasing variation over time. This is particularly obvious if initial size and growth rate are positively correlated. If initial size and growth rate are negatively correlated, then there will be a period of apparent convergence, followed again by the pattern of increasing variation with time. In this case, we are dealing with a series of growth equations, again of the same general form, but allowing for differences in both initial size, and growth rates. If we allow the form of the equation (for example, linear, logarithmic, exponential) to vary among individuals, and perhaps even to vary from time to time within an individual, we can achieve even more complex dynamics, but the mathematics becomes increasingly intractable. Figure 2 presents results involving two developmental variables in yellow cedar, height (fig 2a.) and whorl initiation (fig 2b) (see chapt 2). Rather than tracking individual trajectories, this figure tracks the variation that results from 116 those trajectories. As can be seen, variation among seedlings (Se) increases over time, implying that the developmental trajectories (or unviariate growth curves) of individual seedlings diverge from each other. The previous two graphs consider changes in a single feature with time. However, when we observe plant development, a number of features change with time in a coordinated manner; that is, we must be concerned with not only variances, but also with the nature of the covariances and correlations among plant parts as they change through development. Thus, we will now proceed from a univariate notion of a growth curve (any variable plotted against time) to the notion of a developmental trajectory existing in many dimensions. In this manifestation, the axes represent the variables used to characterize the developing system, the position of an individual at any time can be represented as a point, and time is the ordering principle that connects the points. In this context, I examine three increasingly less restricted models of developmental trajectories. The graphs and examples presented are in two and three dimensions, but the concepts can be generalized to many dimensions. Before proceeding to the models, a brief introduction is given to a number of concepts related to the notion of a developmental trajectory ~ dynamical systems, attractors, phase-spaces, ~ and how they may be related to the concepts of covariance and correlation structures. I also provide a preliminary definition of non-linearity at this point, but leave a more detailed discussion of its implications for later.Further mathematically oriented definitions of nonlinearity are provided in two recent texts on complex systems (Nicolis and Prigogine, 1989; Campbell, 1989). A dynamical system is a system in which the rate of change in each variable is influenced by other variables. Through time, the system changes 117 in its properties (e.g. form, distribution of genotypes, etc. Kauffman, (1989 pp. 627 - 628 ) provides a succinct illustration introducing the concepts of dynamical system, phase space (also referred to as a state space or variable space), trajectory and their relations to each other.: "The most natural language to describe the behavior of an integrated system whose coupled components influence one another's activities is dynamical systems theory. The most familiar form derives from Newton, and consists of a system of differential equations in which the rate of change of each variable is written down in terms of the present values of all variables which influence that variable. To be concrete, suppose there are three chemical variables undergoing reactions in a vessel. The rate of formation and disappearance of each chemical depends upon the concentrations of those chemicals forming it, or influencing its formation, and those influencing its evolution into other chemicals. In addition, each of the three chemicals may be added to or deleted from the vessel, or held constant, or caused to change in arbitrary ways by outside 'forces'. The most natural representation of the construction of such a system is a three dimensional state space where each axis corresponds to the concentration of one of three chemicals. The concentration of all three chemicals at any instant then corresponds to a single point in this three-dimensional space. Over each small interval of time, each chemical may increase or decrease slightly in concentration, due to the reactions represented by the differential equation. Therefore, after one small interval, the representative point will have moved to some (in general) new point in state space. Over a succession of such small intervals, the representative point will move to a succession of (generally) new points. Thus the succession of points may be connected by a smooth line which shows the trajectory of the system across state space, over time, the trajectory itself no longer explicitly shows time increments" In general, for most dynamical systems, the equations governing the evolution of the system are not known, or the initial conditions to which those equations apply to predict the way the system will evolve in phase space are unknown. This is usual in complex, many variable systems such as are encountered in biology and astrodynamics. In these cases however, we can still track the system in a space defined by measured (as opposed to calculated) variables, and reach some general conclusions. The trajectory of a dynamical system is just the line tracking its motion across the state space. In this sense, the state space may be considered the "stage" or "canvass" in which the trajectory acts. The motion of trajectories through 118 such a state space brings us to the concept of attractors. If all trajectories tend towards a certain location in the state space, then that location is called the "attractor" of the trajectories. For example, if all trajectories converge to a single point, then we have a point attractor. In developmental terms, this would be the case if a series of individuals that were initially morphologically different all resulted in the identical mature form. More than one attractor can appear within the phase space. Returning to Waddington's idea of the chreod, or "necessary path", Waddington noted critical points where the manipulation of imaginal discs in fruit flies will switch development towards one mature form or another; i.e. switch from one attractor to another. Attractors can be more complex than single points. A more general definition of an attractor for a dynamical system would be "a set of points or states in state space to which trajectories within some volume of space converge asymptotically over time" (Kauffmann, 1989b pg629). Attractors are said to be "strange" if their dimension is not an integer. For example, if in a two variable space all trajectories are attracted towards a single end point, the dimension of the attractor is 0. If all trajectories end up falling along a line, the dimension of the attractor is 1. If the trajectories are attracted to a line that folds back on itself in a complicated manner, so that it seems to cover area, rather than a line, the attractor would have a dimension between 1 and 2 and be called "strange". It's dimension then is a measure of how much of the total space is occupied. To a morphologist, the terminology in the preceding paragraphs may seem somewhat exotic. However the notion of an "attractor" can be related to the notion of covariance. The fact that variables are correlated with each other means that within a multivariate state-space, certain areas are inaccessible, and that more points are likely in some sub-volumes of the 119 variable space than in others. Thus, as long as there are variable covariances, trajectories are restrained. Covariance structure thus may be seen as representing the constraints on the system (such constraints could be functional, genetic, developmental, physical, or as in the case of coiling in snails (Raup, 1966; Gould, 1980,1989) the interplay between all four factors; covariances identify the possibility of a constraint, they neither guarantee nor explain constraint on their own). Where the system is so tightly constrained that all trajectories converge to a single point attractor, over time correlations approach 1. Terminology of "attractors" implies that with time, the system occupies a smaller and smaller volume of the state space, or variation decreases. However in biological systems, variation often increases through time. This is why covariance among variables is denoted as "restraining" rather than "attracting" trajectories. Systems in which the volume of a point cloud in state space changes with time are called dissipative. Most examples of physical dissipative systems are those where the volume occupied by the point cloud diminishes with time, often resulting from a nonlinear force such as friction. However in biological systems the point cloud often increases in volume occupied with time. Just as the attractors of dynamical systems of equations have a dimension that can be calculated, and represents the volume of the phase space occupied, it is possible to calculate the dimension of a point cloud in variable space, resulting in a measure formally analogous to fractal dimension (see Banerjee et al. for mathematical details) which represents the relative order of the system ~ the more ordered the system, the smaller the proportion of variable space occupied by the point cloud. 120 Finally, having dealt with the basic descriptions of dynamical systems, we should note that most dynamical systems are nonlinear. This means that the trajectories do not fall on a straight line over state space, therefore the trajectories cannot be described by linear equations. Mathematically, the system simply cannot be reduced to a linear function of the input variables, with the "effect" of each variable simply being added to that of the others (this is called the "superposition principle"; in linear systems the parts can be added together, in non-linear systems they cannot). Unlike linear systems, nonlinear systems can display arbitrarily complex behavior over time, show large changes in behavior as the result of small changes in parameters, and display ordered patterns on one scale, where at a lower scale motion seems chaotic and unpredictable (for example, turbulence leading to flow patterns in a liquid; see Campbell, 1989 for other examples). With this background, we can proceed to the models. The first model shows the trajectories to be essentially parallel (fig 3a), the second has trajectories diverging, but in a linear manner (i.e. at a constant rate), and in the third, where trajectories diverge in a nonlinear manner (i.e. the rate of divergence is not constant with time) Using three individuals, we can represent the nature of the correlation structure in a simplistic manner, by drawing a triangle connecting the three individuals at any given time, and then tracking the nature of the changes in the form of the triangle through time. In model 1, (fig. 3a) parallel trajectories, the triangles are constant in both size, form and orientation throughout time, and therefore covariances and correlations would be constant. In the case of diverging trajectories, the size of the triangle increases through time, but the form and orientation of the triangles do not change. This corresponds to 121 increasing variation in which variances - and hence covariances - change through time but the amount and structure of inter-variable correlations remain constant. In the third case of both diverging trajectories and nonlinearity, what occurs is that the triangle changes in both size, shape, and orientation. In this case, concommittant with the increase in variances and covariances, there are changes in the amount and structure of variable correlations. Since the triangle represents the positions of the individuals relative to each other, what this means is that the shape of the individuals relative to each other is also undergoing change. Mathematically, each individual is now represented by a matrix of growth equations, one for each variable measured, and the evolution of that matrix through time can be examined. At a population level,the behaviour of a set of such matrices can be examined. Since the equation for any variable can be of different form, such a scenario, while relatively easy to represent symbolically, becomes extremely difficult to treat computationally, or in a predictive manner. The interesting point is that for (a) variation to increase with time, trajectories must diverge and (b) for correlations to change with time, the divergence itself must be nonlinear, i.e. pairs of trajectories diverge at relatively different rates, so trajectories cannot be superposed upon each other by a simple linear transformation. While the graphs presented are simple, the underlying mathematics is complex, and beyond the scope of this thesis. However, plant development is not concerned with the mathematical intractability of its behavior. Rather than mathematical modelling, we can simply plot out real data and note the graphical model to which they most closely conform. Thus, if we expand our sample from three individuals, to a larger number, we can summarize the distribution of those individuals in 122 terms of 99% confidence ellipses which describe certain properties of the swarm of points. From such ellipses, three statistics may be derived: area, eccentricity, and theta (which is the orientation of the ellipse to some reference axis). These three statistics may be considered simple descriptions of the covariance and correlation properties of the system under study (see, Scagel, et al., 1985). Area represents the overall variability of the system. Eccentricity provides an estimate of the "tightness" of the variable intercorrelations. Where all variables are independent, the ellipse is essentially circular and has an eccentricity of 0. As variable inter-correlations increase, the ellipse becomes increasingly elongated, with an eccentricity of 1 being approached . Changes in theta represent changes in correlation structure or the orientation of the point cloud. It should be noted that the use of 95% confidence ellipses is itself based upon certain model statistical assumptions (Jolicoeur, 1968; Jolicoeur and Mossiman, 1960; Scagel et al., 1985), in which points must approximate a bivariate or multivariate normal distribution; thus this method requires linear relations among variables at each time (so the point swarm may be characterized by an ellipse), though the relations among variables through time can be nonlinear (allowing the ellipses to change in size, form and orientation). In this context, we can extend our previous three models. The case of parallel developmental trajectories leads to a series of ellipses identical in area, eccentricity and orientation (fig. 4) which I designate as a "pure growth model", because through time there are only increases in size, and not shape. Linearly diverging trajectories lead to a series of ellipses identical in eccentricity and orientation, but with increasing area through time (fig. 5), an "increasing variation" model. Finally, if we allow developmental trajectories to diverge non-linearly, we can have changes in 123 both eccentricity and orientation, as the nature of the point-swarm changes, along with increases in area, leading to the "increasing complexity" model (fig. 6). The question is, which of these models corresponds most closely to empirical reality. In the univariate case (fig 2) it was shown that variation among individuals increases with time. In a series of multivariate studies on ovule development in Stipa, Nothofagus, Phyllostachys and two species of Balsamorhiza (Maze et al., 1986, 1987; Scagel et al., 1985; Maze unpublished data) , a multivariate extension of the above described method of confidence ellipses was used and it was found that with time ellipses changed in size, eccentricity and orientation; i.e. according to model three, the data sets increased in complexity with time. Fig 7 presents the results of these studies. In a study of branch growth in Douglas-fir (Banerjee and Maze, 1989) much the same was found, though the changes in ellipse eccentricity and area were less pronounced. This suggests that plant development is often characterized by a phenomenology of nonlinear diverging developmental trajectories which will lead to changes in ellipse areas, orientations, and shape. It should be noted that while the study of Douglas-fir followed the same individuals through time, the ovule study required destructive sampling, and hence the additional assumption that the individuals sampled at each time were representative of the population developing through time. When a pattern of increasing divergence is noted among individuals, the next question that needs to be addressed is whether this divergence is based on genetic differences among individuals or if it is a result of developmental differences caused by differential gene expression. This topic was addressed briefly in chapter two, but since it is pertinent to the following some the 124 discussion is repeated here. The distinction between whether the diverging trajectories are based on genetic or epigenetic phenomena would be as follows. If the diverging developmental trajectories are a reflection of differential gene expression, then we would expect to find a pattern of divergence even when dealing with a population of genetically uniform individuals. The accrual of differences among individuals with ontogeny would then be a reflection of the differential expression of common information in the course of ontogeny. In this respect, it should be noted that in the Phyllostachys and Nothofagus data sets, ovules were derived from a single plant, and still diverged with time. In the Stipa data sets, ovules were gathered from three different individuals, and while there was divergence within each individual (increasing ellipse area) there was also divergence among the two individuals (separation of ellipse means). What this implies is that development alone is sufficient to lead to diverging trajectories even when individuals are genetically uniform, and that this trend can be further amplified by genetic differences among individuals. In the data presented in chapter two, we noted a pattern of increasing variation among individuals within families through time . This implies that the trajectories (or in this case, growth curves) of the individual plants are diverging with time. However, given when individuals are derived from open-pollinated families, genetic differences should be expected, and thus it is not possible to determine to what extent epigenetic and genetic factors contribute to these differences. Such an assessment could be accomplished in the context of work on pure lines or clonal material where comparisons could be made by comparing the rate of divergence in developmental trajectories for individuals with a common genotype versus individuals with different genotypes. 125 In summary, the development of an individual can be viewed as a dynamical system, and such a view leads to a particular analytical approach to studying development and interpreting developmental sequences, or their gross manifestation as changes in morphology. 126 Development as a Complex System: Models, Math and Reality John Maynard Smith ends the introductory chapter to his book "Evolution and the Theory of Games" with a brief consideration of theory, models, and their connection to empirical data that is worth repeating: (Maynard Smith, 1982:pg 9) "I think it would be a mistake, however, to stick too rigidly to the criterion of falsifiability when judging theories in population biology. For example, Volterra's equations for the dynamics of a predator and prey species are hardly falsifiable. In a sense they are manifestly false, since they make no allowance for age structure, for spatial distribution, or for many other necessary features of real situations, their merit is to show that even the simplest possible model of such an interaction leads to sustained oscillation - a conclusion it would have been hard to reach by purely verbal reasoning. If, however, one were to apply this idea in a particular case, and propose, for example, that the oscillations in numbers of Canadian fur-bearing mammals is driven by the interactions between hare and lynx, that would be an empirically falsifiable hypothesis. Thus there is a contrast between simple models, which are not testable, but which may be of heuristic value, and applications of those models to the real world, when testability is an essential requirement". In a similar manner, I have tried to present three general models of development and then ask which model corresponds to the general patterns found in developmental studies. However, the specific mechanism by which the developmental trajectories occur has not been dealt with. Thus, a model specific to, for example, the development of individuals in Stipa would 127 require detailed knowledge concerning the mechanisms involved in triggering sequences of cell division. In the absence of such information, we have been forced to "black-box" large areas. A similar form of "black-box" is found in quantitative genetics, where for the sake of simplicity it is argued that traits of interest are governed by the action of numerous genes with tiny additive effects and no epistatic effects. Similarly, the increasing complexity model assumes some form of nonlinearity, but signifies little about the specific nature of the nonlinearity. It does however guide us to observe those cellular processes that can act iteratively, since iterative processes are often a source of non-linearity. In this case the focus would be directed to determining how regulatory genes initiate and direct cell divisions. In what follows, I wish to expand on the degree to which models in complex system research have guided intuition, and the degree to which such models depart from, or over simplify reality. When considering examples of complex systems, the images that come to mind are often biological. People contemplate the way a zillion different neurons manage to work together every moment we think, or even when we're not thinking. They ponder how a single cell multiplies and differentiates in the course of development. They regard systems with many different parts all linked together and working coherently. And yet, oddly enough, much of our intuition concerning the nature of complex systems comes from the study of an extremely simple equation, the logistic difference equation (May, 1976; May and Oster, 1976), that models certain aspects of how a population ~ say fruit flies or bacteria in a jar, or lynx in the forests of northern Ontario ~ might change with time. In this equation, the size of 128 the population at any time depends on its size at a previous time, and a "tuning parameter" called "r" which is the intrinsic rate of increase: X t + i = Xt-*r(l- X t) X = population size t= discrete time r = intrinsic rate of increase K r < 4 This equation is not very long, contains few parameters, requires no knowledge of calculus for solution, is overall not very intimidating in appearance. So what is special about it, and how can it tie into the complex types of behavior, the mix of predictability and unpredictability we associate with biology? In response to this: the equation is nonlinear. Indeed, the higher you set the "tuning" parameter, "r", it becomes increasingly nonlinear. At a certain "r" value (3.57), the equation becomes "chaotic". That is, rather than the population increasing steadily with time, it fluctuates up and down in no predictable manner. In the long run, we can not guess where the population will end up. We have, therefore, the results of a simple, totally determined equation, leading to unpredictable behavior. This diversity out of simplicity seems analagous to the fact mentioned earlier that the vast array of plant morphologies arise out of a relatively minor catalogue of cellular events concerning the plane and frequency of division and elongation. What else is interesting about the equation is that two populations can start at almost the same initial population size, and with time they will begin to diverge. The pattern of change in one population will become increasingly different from the pattern of change in the other population, even though they are governed by exactly the same rules. This is called "sensitive dependance on initial conditions" and is due to 129 the fact that small differences in initial conditions are exponentially multiplied over time; i.e. trajectories diverge even when they originated arbitrarily close to each other. The implication is that while initially, knowing the state of one population (or for a more developmentally oriented example, a genetically identical clone in a uniform environment) allows accurate predictions concerning the state of the other, as time goes on, knowing the state of one population leads to increasingly unreliable predictions concerning the state of the other population (or clone). As a model to be tested against empirical data concerning patterns of change through time, the logistic difference is wholly inadequate. Indeed, since the range of X values (which represent population size) in which the model will not proceed to negative infinity lies between 0 and 1, the equation would seem more a mathematical curiosity, than something of relevance to field or experimental biologists seeking to quantify data gathered on population or developmental dynamics. Its value lies in the fact that the equation seems, on a qualitative level, to capture certain aspects of the more complex biological systems that we're really interested in. The logistic difference is arguably the simplest of nonlinear equations with a simple feedback rule, which exhibits complex behavior due to its intrinsic dynamics (as opposed to external perturbations), including a simple form of "memory", dependance on initial conditions, a range of parameters where stable solutions result and trajectories converge (e.g. homeostasis), and beyond which "chaotic" behavior occurs leading to multiple solutions, bifurcations, areas of apparently random behavior, and surprisingly, areas of order. As such, the logistic difference equation is suggestive of the direction more elaborate biological models 130 may take. What I have argued throughout this chapter is that for developmental trajectories to lead to a pattern of increasing variation, changing covariances and correlation structures with time, the underlying processes must be nonlinear. Such changes in correlation structure are not only characteristic of the development of individuals, but also of the morphological differences among species (Robson,1989; Robson et al., 1988). While the present discussion has concerned primarily the relationship between developmental trajectories and the emergence of within population variation, the extension of similar arguments to the level of species remains an interesting possibility. Whether any single equation, no matter how complex, can actually capture the empirical details of the development of a population of individuals is an open question whose elucidation will require much work. 1 3 1 References Alberch, P. 1980. Ontogenesis and morphological diversification. Amer. Zoo. 20:653-667. Alberch, P. 1985. Problems with the interpretation of developmental sequences. Sys. Zool. 34:46-58 Alberch, P., S.J. Gould, G.F. Oster, and D.B. Wake. 1979. Size and shape in ontogeny and phylogeny. Paleobiology 5:296-317 Atchley, W.R. 1987. Developmental quantitative genetics and the evolution of ontogenies. Evolution 41:316-330 Atchley, W.R. and Newman. 1989. A quantitative-genetics perspective on mammalian development. Am. Nat. 134:486-512 Banerjee, S. and J. Maze. 1988. Variation in growth within and among families of Douglas-fir through a single season. Can. J. Bot. 66:2452-2458. Banerjee, S., P.R. Sibbald, and J. Maze. 1990. Quantifying the Dynamics of Order and Organization. J. Theor. Biol. 143:91-112 Campbell, D.K. Nonlinear Science: from paradigms to practicalities. In, "From Cardinals to Chaos. Reflections on the life and legacy of Stanislaw Ulam" Edited by N.G. Cooper. pps:218-262 Creighton, G.K. and R.E. Strauss. 1986. Comparative patterns of growth and development in critocene rodents and the evolution of ontogeny. Evolution 40:94-106 Gleick, J. 1987. Chaos: making a new science. Viking, New York. Gould, S.J. 1977. Ontogeny and Phylogeny. Harvard Univ. Press. Cambridge, Mass. Gould, S.J. 1980. The evolutionary biology of constraint. Daedalus. 109:39-52. Gould, S.J. 1989.A developmental constraint in Cerion, with comments on the definition and interpretation of constraint in evolution. Evolution. 43:516-539. Holland, J.H. 1989. Using classifier systems to study adaptive nonlinear networks. In, "Lectures in the sciences of complexity". Edited by D.L. Stein. Addison-Wesley Publishing Co. pp463-499. Jolicouer, P. 1968. Interval estimation of the slope of the major axis of a bivariate normal distribution in the case of a small sample. Biometrics 49:679-682 132 Jolicouer, P., and J. Mossiman. 1960. Size and shape in the painted turtle. A principal component analysis. Growth 24:339-354 Kaplan, D.R. 1975. Comparative developmental evaluation of the morphology of unifacial leaves in monocotyledons. Bot. Jahrb. Syst. Pflanzengesch. Pflanzengeogr. 95:1-1-5 Kauffman, S. 1989a. Adaptation on rugged fitness landscapes. In, "Lectures in the sciences of complexity". Edited by D.L. Stein. 1989. Addison-Wesley Publishing Co. pp527-618. Kauffman, S. 1989b. Principles of Adaptation in complex systems. In, "Lectures in the sciences of complexity". Edited by D.L. Stein. 1989. Addison-Wesley Publishing Co. pp619-712. Kluge, A.G. 1988. The Characterization of Ontogeny. 1988. In, "Ontogeny and Systematics". Edited by, C.J. Humphries. pp:57-81 Kluge, A.G. and R.G. Strauss. 1985. Ontogeny and Systematics. Ann. Rev. Ecol. Syst. 16:247-268 May, R. 1976. Simple mathematical models with very complicated dynamics. Nature 261:459-467 May, R. and G.F. Oster. 1976. Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110:573-599 Maynard Smith, J. 1982. Evolution and the theory of games. Cambridge University Press. Maze, J., R.K. Scagel and L.R. Bohm. Quantitative studies in early ovule development. II. Intra- and inter-individual variation in Stipa lemmonii. Can. J. Bot. 64:510-515 Maze, J. R.K. Scagel and L.R. Bohm. 1987. Quantitative studies in early ovule development. III. An estimate of shape changes in Phyllostachys aurea. Can. J. Bot. 65:1531-1538. Maze, J., S. Banerjee and Y.A. El-Kassaby. 1989. Variation in growth rate within and among full-sib families of Douglas fir. {Pseudotsuga menziesii) Can. J. Bot. 67:140-145 Maze, J., K.A. Robson, S. Banerjee, L.R. Bohm and R.K. Scagel. 1990. Quantitative studies in early ovule and fruit development: developmental constraints in Balsamorhiza sagittata and B. hookeri. Submitted. Mittenthal, J.E. 1989. Physical aspects of the organization of development. In, "Lectures in the Sciences of complexity". Edited by D.L. Stein. pp225-274. 133 Namkoong, G. and M.B. Conkle. 1976. Time trends in genetic control of height growth in Pinus ponderosa. For. Sci. 22:1-12 Nicolis, G. and I. Prigogine. 1989. Exploring complexity: an introduction. W.H. Freeman and Co., New York. Raup. 1966. Geometric analysis of shell coiling: general problems. J. Paleotol. 40:1178-1190. Robson. K.A. 1989. Phylogeny, Ontogeny and Intraspecific Variation in Balsamorhiza. (Asteraceae, Heliantheae): a study of evolutionary phenomena. Phd. Dissertation. University of Nebraska: Lincoln, Nebraska. Robson, K.A., R.K. Scagel and J. Maze. 1988. Within-species organization in Wyethia and Balsamorhiza and an assessment of evolutionary explanations. Taxon. 37:282-291 Scagel, R.K., J. Maze, L.R. Bohm and N.L. Vogt. 1985. Quantitative studies in early ovule development. I. Intraindividual variation in Nothofagus antarctica. Can. J. Bot. 63:1753-1778. Waddington, C.H. 1957. The strategy of the genes. Allen and Unwin. London. Waddington, C.H. 1977. Tools for Thought. Jonathqn Cape Thirty Belford Square London. 134 Figure 4.1: Two examples of growth curves. The vertical axis is the measured variable, the horizontal is time. In (a) all growth curves are parallel, in (b) the curves diverge from each other. V a ) Time 135 Figure 4.2: The apportionment of variation for (a) height (Ht) and (b) whorl (Wh) data. The vertical axis is the square root of the sums of squares from a nested ANOVA, the horizontal is time. To = total variation, Si = variation due to sites, Fa = variation due to open-pollinated families, Se= variation due to seedlings. T i m e 136 Figure 4.3: Three models of developmental trajectories. In (a) trajectories are parallel, in (b) trajectories diverge, in (c) trajectories diverge non-linearly. All three axes represent variables, and the trajectories occur in a multivariate space. 137 138 Figure 4.6: Ellipsoids of variation corresponding to the "increasing complexity model" where developmental trajectories diverge nonlinearly 140 Figure 4.7: Major axes of ellipsoids of variation from developmental studies of (a) Stipa, (b) Phyllostachys, (c) Nothofagus and (d) Balsamorhiza. This figure is provided courtesy of J. Maze. STIPA a ) PHYLLOSTACHYS b) NOTHOFAGUS c ) BALSAMORHIZA d ) THE ONTOGENY OF MORPHOLOGICAL VARIATION: AN EXAMPLE FROM YELLOW-CEDAR Epilogue: Where to From Here, or, What's It all Mean Mr. Natural? Satindranath Banerjee Department of Botany University of British Columbia Vancouver, B.C. V6T 2B1 142 This thesis presents a series of studies — empirical and theoretical — which attempt to integrate developmental biology with population level studies of variation to initiate a "developmental population biology" that would complement the well established fields of population ecology and population genetics. The creation of such a developmental population biology is important for theoretical reasons in that it provides a link between the development of individuals and variation in natural populations. It is patterns of variation in natural populations that are central to most evolutionary theorizing. Furthermore, from a practical point of view, it is the variation in natural populations that is manipulated by forestry practices and the approaches detailed in the thesis allow for a deeper understanding of the sources of that variation. The results of the thesis center around a series of studies conducted on yellow cedar, a high elevation conifer of some economic importance. One study examines seedling growth variation and its interdependence on parentage, developmental history and local environment which interact in complex ways to structure variation in seedlings. Another study examines time trends in seedling growth demonstrating that most of the increase in variability with time results from seedlings within a family becoming different from each other. A final study examines the nature of variation within individuals, demonstrating that the structure of such variation changes with development. These results have strong silvicultural applications, particularly with respect to wild stand seed collections, growth of seedlings in a nursery, and clonal propagation which seeks to manipulate variation within individuals. The results of the empirical studies are placed into a theoretical context, through the development of a series of models (based on dynamical systems 143 theory) of developmental variation. These models link the development of individuals to specific patterns of variation at the population level. This is the first step required in creating a "developmental population biology". In attempting to integrate development with population level studies, I have examined development from a perspective where the analytical tools of population biology may be applied: hence my concentration on the notion of a developmental trajectory. Such a trajectory may be seen as a summary (potentially quantifiable) of the events occurring in the ontogeny of an individual. A series of such trajectories summarizes the variation occurring with development in a population. The mathematical concepts for developmental trajectories were developed in the context of physical systems — particularly statistical mechanics. In the process, I am creating metaphorical links among a number of fields in biology and physics: qualitative studies of development, population dynamics, dynamical systems in general. The disparate nature of these fields, each with its unique history, guiding thema, and independant terminology, makes communication difficult. To ease the translation between intellectual areas I have tried to be careful to define my terms and introduce the concepts gradually. Thus, through the thesis the ratio of empirical data to theoretical considerations gradually shifts. The first two chapters rely primarily on simple graphical summaries of medians and variation about the median. The last two chapters deal with more abstract forms of analysis. Hopefully, in this fashion, the notion of developmental trajectory gradually becomes increasingly comprehensive. It is an abstract concept that is rooted in biological data but can be extended to a physical interpretation. A tie to the physical world allows the discussion of development to be placed in a more general framework. The price of the generality however, is 144 abstraction, the loss of certain elements of fine detail, that are the stock and trade of the observational biologist. It has been said that the distinction between a theorist and an empiricist is that no one believes a theoretical work but its author, whereas everyone believes an empirical work except its author. To combine theory and empiricism is to risk having no-one believe your results including yourself. Yet, I do believe in my results. Why? I do not really know. Part of it has to do with the familiarity one gains with the objects of study in the course of a research project. Second, in the course of measurement and analysis, the relationship between qualitative observations, and analytical results becomes clearer, as does the utility and limitations of conducting your analysis according to certain theoretical pre-conceptions. For example, the calculation of the determinant of a correlation matrix may be seen as a way of solving several linear equations simultaneously, or it may be seen as a method for creating a numerical summary of the variation in a data set. Each interpretation of the determinant will affect how one analyzes the data, and indeed, the results obtained therefrom. If I were to assess the degree of success I have had in integrating theory and data, my answer would depend upon whether the criteria for success were conceptual, analytical or empirical. From the point of view of conceptual integrity, I believe I have provided a unified series of ideas (an amalgam of the work of many other people) which allow development to be placed in a more general framework. From the point of analytical sophistication, I would deem my success to be moderate. Analytically, I have used very rough tools, attempting to study non-linear systems using analytical protocols developed in linear algebra. I have gotten around this by presenting the nonlinear patterns primarily graphically. The nonhnearity 145 refers to the relation among individuals through time, whereas the relation among individuals at a given time do correspond to the assumptions required for the analyses I use. These limitations are perhaps due jointly to my own analytical background and limitations, and the fact that non-linear problems are often mathematically intractable. A mathematician would not be very satisfied with my tools, but then, he would not have known how or what to apply them to either. Finally, from the point of view of an empiricist, I would have to judge my success to be relatively modest. I have spoken of developmental trajectories as representing, at the histological level, a sequence of specific events relating to the plane, position, and timing of cell divisions and elongations. However nowhere in this study have I directly assessed these phenomena. If I had known a priori the direction my thesis was to take, I probably would have chosen a different organism to work on, the simplest multicellular plant I could find, rather than one of the most complex. Hopefully I will be able to rectify the empirical weakness of this study in future studies where I may examine the notion of developmental trajectories in the context of data sets where information is gathered concerning the plane, position and timing of cell division and elongation for numerous individuals. Based on these criteria, my thesis can be judged as a rather ambitious failure. That is an assessment I am quite comfortable with. Most new scientific ideas are (a) not so new or (b) not so true, and often both. However, as long as new ideas keep being generated, eventually a few will be neither "a", nor "b". The commitment to developing ~ either conceptually, analytically, or empirically - certain theoretical ideas in science is an exercise in risk. For example, consider John Wheeler (a physicist) on the Everett-Wheeler interpretation of quantum theory: 146 "I confess that I have reluctantly had to give up my support of that point of view in the end - much as I advocated it in the beginning » because I am afraid it creates too great a load of metaphysical baggage to carry along. But they say nobody knows sin like a sinner." Yes, and in science, the road to hell is paved with grand unified theories, as is the road to heaven. 

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