p19. An autopoietic unit is a system that determines its own making, due to a network of reactions which take place within its own well-defined boundary.
p20. It follows that autopoiesis cannot be ascribed to any single component, but rather to the operational unity of the whole system, echoing the intuition that life cannot be ascribed to any single molecular component (not even to DNA or RNA!) but only to the entire bounded metabolic network. It is apparent at this point that the notion of autopoiesis has its matrix in biology, but that it reaches into the more general issues raised by the theory of complex systems. Note also that it is the organization of the system which is stressed: structure (i.e., the actual material components) is almost secondary, in the sense that an autopoietic unity can be realized with several different specific structures.
p21. closure refers in general to the containment of system operations within a system boundary: in living systems, closure means that the ... system is materially and energetically open, but operationally closed.
p41. In biology the property of returning to the state a system was in before it was disturbed is calleds homeostasis .... C. H. Waddington ... introduced the words homeorhesis (similar flow) to describe a system which returns to a trajectory and chreod (necessary path) for the trajectory itself. He also used the term canalization to describe the property that development typically can proceed to one or more of a restricted number of alternative end states rather than to a broad spectrum.
p46. we can define a dynamical system as a manifold with a vector field defined on it. In other words, we describe a dynamical system by specifying the set of possible states, the phase space, and then providing rules that tell complete us at each point in the space where to go next. The rules can be given as differential equations ..., difference equations or cellular automata.
... the epigenetic landscape represents a class of dynamical systems which share a number of important properties which are typically found in developmental systems. The mathematical problem is to determine the properties of this class.
p48. Relatively small changes in parameters can, by causing attractors to disappear, bring about large changes in the state of the system. Thus large effects do not have to have large causes, as they generally do in linear systems. What is more, in nonlinear systems different causes can have the same effect. In particular, when a large change is possible, it can usually be initiated by altering the value of any one of a number of different parameters. This is very important for evolution, because it means that not only are large changes possible, they do not depend on one particular mutation nor even on a small set of mutations, each with very much the same effect.
p49. two dimensions is a very special case in dynamics. This is largely because we are concerned with trajectories, which are one-dimensional, and there are many things that are true only when the dimension of what you are studying is precisely one less than the space that it is in. ...
... It does not matter very much how many dimensions there are, as long as it's more than two. Another peculiarity of differential equations in two dimensions, incidentally, is that the only attractors are points and limit cycles. In three or more dimensions, there can also be strange attractors, i.e., chaos.
p59. some regions of the phase space which correspond to permitted values of the parameters are for the most part inaccessible, because they can only be reached from normal starting points by passing through a forbidden region. In effect, the phase space for the real process may not be connected. It might be, however, that a perturbation could force the system into such a region. This would correspond to forms that can occur only through environmental disturbance, never by normal development or mutation.
p60. while the assumption that a phase space is a smooth, simply connected manifold is seldom stated explicitly, many results depend on it. For example, a number of authors have discussed evolution in terms of fitness landscapes with species striving to reach adaptive peaks, i.e., phenotypes with (locally) maximum fitness. In principle there is no reason why one cannot imagine a phase space for phenotypes. On the other hand, its topology ... would be very different from that of a Euclidean space. At any point we can move only in a very restricted number of directions, and we cannot go from almost any point to almost any other point by a continuous curve. ... That there can be large phenotypic changes implies that there are points in the phase space that are in one sense far apart and yet in another sense close together. We should not assume that just because it is possible to define a space, it, and therefore the system or process it is supposed to model, will have all the nice properties of the Euclidean plane.
p62. we often study a particular organism more because of the light it can throw on general problems than for its own sake. The aim of the vast amount of research that has been done on Drosophila has been to learn about genetics and development, not to satisfy an apparently limitless curiosity about fruit flies.
... we can also use particular dynamical systems as examples to help us in our work. [These] are what we might call mathematical fruit flies. We are interested in them not because they model important chemical reactions---it may even be that they do not model any real reactions at all---but because they are convenient to work with and yet share important properties with large classes of dynamical systems which almost certainly include many of those that occur in development.
p106. Systems built of parts with low connectivity (a given part interacts with a small number of other parts, very often neighboring ones) and with no dynamical conflicts (it is possible to optimize the system with respect to all the different restrictions) have, in general, one or just a small number of optimal states and the system arrives at them, independent of the initial conditions and perturbations suffered along its history. Field descriptions of morphogenesis usually belong to this class. On the other hand, systems with large connectivity and "frustration" (conflicts among the different optimization constraints) have, in general, a large number of quasi-equivalent states of almost equal optimality. If the connectivity is large enough, these states are separated by regions of low optimality and the particular state reached by the system depends strongly on the initial conditions and on its previous history. Immunological and neural networks are examples of this last type. Therefore, depending on the degree of connectivity and conflictness, we can move from an "analytical" to a "historical" system and we can consider cases where systems of different type coevolve.
p132. The process is known mathematically as a moving boundary problem: the dynamics of the ... cell wall system generates a geometrical form which then acts back on the dynamics which, in turn, generates a new geometry. Not much is known about such systems, but they are the category of process to which developing organisms belong.
p136. there is a clear research programme for the study of biological morphologies as natural forms, as attractors in the space of morphogenetic field dynamics.
p139. More than 80% of plant species have spiral phyllotaxis, but there is no correlation between these and habitat regarding light or shade preference. The explanation for the high frequency of spiral phyllotaxis is, in all probability, simply the greater robustness and stability of this pattern (i.e., a greater domain of attraction of this pattern in the parameter space of the meristem as a developmental generator). However, this conjecture remains to be further studied.
p141. Organism and environment together define the developmental dynamic and morphogenetic trajectory.
p143. The processes involved [in forming the vertebrate eye] are robust, high-probability spatial transformations of developing tissues, not highly improbable states that depend upon a precise specification of parameter values (a specific genetic program). The latter is described by a fitness landscape with a narrow peak, corresponding to a functional eye, in a large space of possible nonfunctional (low fitness) forms. Such a system is not robust: the fitness peak will tend to melt under random genetic mutation, natural selection being too weak a force to stabilize a genetic program that guides morphogenesis to an improbable functional goal. The alternative is to propose that there is a large attractor (a large range of parameter values) in morphogenetic space that results in a functional visual system-i.e., eyes have arisen independently many times in evolution because they are natural, robust results of morphogenetic processes.
p144. organismic morphologies are the result of robust morphogenetic processes, evolution is the emergence of the generic forms of the living process. Characterizing these generic forms will then provide us with the principles that can make sense of both the regularity and the diversity of the biological realm, just as characterizing the generic modes of any dynamical system explains both its order (regularity) and the variety of expression of this order. ... The problem is always to discover the generative source of biological order, the structuralist foundations that are logically prior to the functionalist and historical preoccupations of neo-Darwinism.
p151. ... concept of a uniform determinism which ... has moulded the basis of classical science. According to this concept, each natural event should have its own specific cause which should be of the same order of magnitude. This implies that negligibly "small" causes cannot generate noticeable events, and vice versa. One of the main postulates of uniform determinism is also known as the "one cause-one effect" rule. ... this rule has often been taken for granted and widely used, although mostly in an intuitive and undefined way. But is such an approach really adequate for developmental processes so often associated with symmetry breaking?
p153. These systems do not obey … uniform determinism. They are capable of "spontaneous" dissymmetrisation; negligibly small causes may produce in these systems enormously large effects; one "cause" can generate more than one effect; some strictly localized "causes" may produce delocalized effects, and vice versa. Generally, any kind of distinction into discrete "causes" and "effects" looks inadequate for this class of systems. They should be largely replaced by quite other notions, such as "stability" and "instability" (it is only instability which permits the "spontaneous" dissymmetrisation), parametric and dynamic influences, and positive and/or negative feedback. …
… Instead of trying to discover some unique and specific cause for each developmental process, we should now focus our interest on revealing the fundamental nonlinear feedbacks which underlie the entire flow of the developmental processes and lead, in some of its space-time areas, to loss of stability.
p155. self-organization means the creation of macroscopical patterns by the action of forces distributed in a much more homogeneous way than the structures that arise.
p156. mechanical ways of self-organization have some advantages over chemokinetical ones. ... It would be very strange if organisms, already in early evolution, did not use these almost unavoidable and straightforward mechanisms for regulating their shaping.
p166. generic shapes should often appear within the course of evolution even if they have no adaptive values and are not supported by natural selection. ... nongeneric shapes may be expected only if they have some real adaptive value allowing the selection of the specific combinations of parameters
p167. is a developing organism no more than a container for the genes which operate as sole masters by their own rules and which change from time to time the container's shape, or is it an integrated multileveled system, each level possessing its own self-organizational properties and being able to affect the others?