if the turbulent nature of the flow is examined in detail, we find that we need to use probability distributions to describe the velocity, and even that is incomplete. We use the simplified abstraction so often, and with such success, that we often forget the underlying complexity.
In biology, the uncertainties behind the abstractions are often closer to the surface. For example, consider the concept of species. We would like to define species such that all living things belong unambiguously to a species. This turns out to be impossible. One definition is that a species is a group of organisms that interbreed with each other in nature and produce healthy and fertile offspring. Thus, the horse and the donkey are different species even though they can mate, because all their offspring are infertile. A number of problems occur with this definition. First, not all organisms reproduce by breeding (sexual reproduction). More important for this discussion is that there are populations we would like to define into separate species that can interbreed, such as the domestic dog and the African golden jackal. So perhaps we would alter the definition to include organisms that can potentially interbreed. However, there are cases in nature where organism A can breed with B, B with C, C with D, but D cannot breed with A.
Similar problems occur every time we make a classification. The euglena is a single-celled organism that can move at will through its aqueous environment. This motility, together with its lack of a cell wall, would indicate that it should be classified as an animal. However, it has the green pigment chlorophyll, which enables it, like a plant, to capture light energy. Biologists have created a separate category, the protists, in part to eliminate the problem of where to put the euglena. However, some protists, the algae, are very similar to plants; others, such as protozoans, are animal-like. Thus, the classifications lack an iron-clad quality. Textbook definitions sometimes include the word mostly, as in ''animals are mostly multicellular."
Whether an organism is single-celled or multicellular is an important characteristic used in classification. However, some may either be at different stages in their life cycle or may simply change in response to environmental conditions. The slime mold is an unusual organism that grows on the forest floor and behaves at one stage as a mass of single-celled protozoans; at another stage the cells fuse into a single supercell with many nuclei; and at yet another stage it forms fruiting bodies on stalks resembling a plant.
Another idea that must be recognized as somewhat arbitrary is the notion of an event. This is another kind of useful fiction. In classical science (i.e., other than in quantum mechanics), there are no events, only processes. Consider the ''moment of conception," when a sperm enters and fertilizes an egg. Examined more closely, we must realize that the event consists of a sequence of changes. If we say that fertilization occurs as soon as a sperm penetrates an egg's cell membrane, we must ask, "penetrates how far?" If, instead, we place the event at the moment that the chromosomes join into a single nucleus, we must ask how complete the joining must be. It is like asking when two asymptotic curves combine. The problem is not that science cannot say when the critical moment occurs. No such moment actually exists.
Another thing that is important to appreciate about biology is that a certain amount of caution is necessary when making predictions or judgments about the validity of reported observations. Living things often surprise and contradict. The following quote by the Dutch biologist C. J. Brejer (1958) applies as well to biology as a whole: ''The insect world is nature's most astonishing phenomenon. Nothing is impossible to it; the most improbable things commonly occur there. One who penetrates deeply into its mysteries is continually breathless with wonder. He knows that anything can happen, and that the completely impossible often does.''
There is no perfect way around these difficulties; the complexity of nature sometimes resists our attempts to coordinate, arrange, and systematize it. All life-forms are unique, defying easy classification; so we create a working definition and go on. We must use the distinctions when they are useful and alter them when they're not.
Biology is also unique in the number of levels of scale that it is necessary to examine in its understanding. In Section 2.2 we describe how living systems can be examined at many levels of detail, from the chemical to the cell to the organism to the ecosystem. Within each level are numerous types of entities (e.g., cells or organisms) and myriad instances of each type. The number of potential interactions is staggering. Sometimes the reductionist approach is appropriate and an individual entity or interaction will be studied almost in isolation. Other times it is necessary to look holistically at the behavior of the group.
It is often the goal of scientific studies to "explain" behaviors observed at one level by looking at behaviors of its component parts. For example, the primary productivity (production of algal biomass) of a eutrophic lake can be predicted by measuring the productivity of individual species cultured in a lab under similar conditions. However, it is often the case that the aggregate behavior of numerous individuals cannot be predicted straightforwardly, even if the behavior of the individuals were well understood. For example, proteins are polymers of 20 different amino acids, in varying sequences. Although the individual amino acids do not function as catalysts, proteins do. A protein is not simply the "sum of its parts.'' "New" properties that arise from the interaction of numerous similar parts are called emergent properties. A mathematical field of study called complexity theory has arisen to study the relation between large numbers of interacting autonomous parts and resulting emergent properties.
A related source of complexity is chaotic behavior. Chaos is dynamic behavior characterized by "extreme sensitivity to initial conditions.'' Consider the sequence of real numbers, (xj, x2,...) between 0.0 and 1.0 generated by what is known as the quadratic iterator:
Figure 1.1 shows plots of two such series that start at nearly the same point, 0.9000 and 0.9001, plus a plot of the difference between the two series. Note that after a dozen iterations the two series become uncorrelated. The difference between the two series seems almost random. This demonstrates that although we know the rule generating the data, without perfect knowledge of the initial condition, we cannot predict very far into the future. This extreme sensitivity to initial conditions is also called the butterfly effect, after the analogy for chaos in weather systems which states that a butterfly flapping its wings in Beijing can cause a hurricane in New York City two weeks later.
A consequence of chaotic behavior is that extremely complex behaviors can result from very simple rules, as in the example just given. The paleontologist Steven Jay Gould proposed that small mutations can greatly alter body plans, producing great leaps in evolution. The benefit of this for the study of biology is that complex processes do not rule out the possibility of simple explanations. The difficulty is that it places a limit on the reductionist view. Having a high degree of understanding of the dynamics of nerve cells does little to explain how the human brain can so quickly recognize a face or decide on a chess move.
a. Initial point = 0.9000
a. Initial point = 0.9000
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