Epistemological Intimacy: A Move to Autoethnography

In this article, the author reflects on the dilemma she faced when choosing an appropriate qualitative method for her master's thesis, which is entitled Creatively Rehabilitating Self-Esteem After an Acquired Brain Injury: An Auto-Ethnography of Healing. She found herself in a unique “insider” position, because, as well as being the student researcher, she was from the same culture as the participants. Therefore, to gain maximum access to the valuable data available, the author chose also to be a participant in the study. She chronicles her journey while choosing the most suitable method. The study, which was conducted as a requirement of her master's program, was eventually completed as an autoethnography.

Introduction
Where should I place myself? Who was I in the study? Was I using the most appropriate method to unearth the rich data I believed were present, data that I needed best to answer the research questions? I considered these questions as I conscientiously collected data and followed standard university procedure during my first foray into qualitative research. I was intent on completing a study and writing a thesis, requirements for the master's in education program in which I was enrolled. The result, which was completed after a journey of relentless methodological questioning that was at times too personal, is a thesis called Creatively Rehabilitating Self-Esteem After an Acquired Brain Injury: An Auto-Ethnography of Healing (Smith, 2004).
In 1997, I sustained a severe acquired brain injury (ABI). After 3 years of recovery, I shut the door on my preinjury career and went back to school. For my master's thesis, I chose to explore the impact of creativity on the self-esteem of individuals who have sustained ABIs from the perspective of the unique culture of ABI in which I found myself postinjury. I felt privileged to be able to offer such an intimate and exclusive "insider" portrayal of life after an ABI and the struggle that such a life can be. My experiences engaging in creativity during my recovery from ABI provided the inspirational fuel I needed to decide what to explore for my thesis. How did I arrive at the method I eventually chose to use for the study? How did I become a participant as well as the researcher? These questions address issues that make interesting stories of process, stories that I will tell after giving a summary of the study.

The study
The study involved 4 participants and included me as the researcher and 5th participant. The purpose of the study was to explore the use of creative activities to enhance the self-esteem of individuals who have sustained an acquired brain injury. The participants were clients of a local head injury rehabilitation center. The therapists at this facility recommended individuals who they felt would be beneficial to the study and would also be affected positively by being part of the study. All of the participants signed ethics consent forms, and they were all offered copies of the completed thesis.
During the 6-week period of data collection, each participant completed a creative project. Data were collected in multiple forms: participant observation (Marshall & Rossman, 1999;Tedlock, 1991Tedlock, , 2000, informal interviews as suggested by Creswell (1998), conversations, group informal interviews, e-mails, and talks with third-party therapists. The data analysis phase began with the transcription of the interviews and member checking of these interviews. I then coded the data by hand, making full use of the multiple forms of data I had collected. Triangulation of these data sources legitimized the findings (Creswell, 2002;Merriam, 2001;Wolcott, 1994). After my "prolonged engagement with the data" (Marshall & Rossman, 1999, p. 154), I eventually uncovered four main themes. The findings indicated that engagement in creative activities is a positive addition to ABI rehabilitation because of its favorable impact on self-esteem.
My personal experience during my rehabilitation had shown me that completing creative projects had a significant impact on my self-esteem. During recovery from any ABI, the focus is totally on therapy and rehabilitation. In the first 2 years of my rehabilitation, I had no short-range goals, projects that provide a sense of accomplishment, pride, and increased self-esteem to any individual. The first goals I had that had conveyed these needed feelings consisted of the completion of creative projects. When I returned to school, I felt that the use of creative activities to enhance the self-esteem of individuals who had sustained ABIs was not only something that I could investigate but also something to which I could contribute personally.
Other researchers have explored similar topics, ranging from the use of creative activities to improve self-esteem and sense of self in individuals, to using creative activities as an alternative form of communicating feelings and thoughts (Aldridge, 1991;Gamwell, 2002;Henderson & Gladding, 1998;Jennings & Minde, 1993;Johnson, 1984Johnson, , 1985Johnson, , 1999McNiff, 1997;Tamminen, 1998). The positive impact of engaging in creative activities on self-esteem in individuals who have disability or are ill (Gladding, 1995;Sacks, 1986;Simon, 1997) has also been explored. This study takes one further step, because it examines the consequences of engaging in creative activities when the participants are survivors of ABIs. Certainly, the study is unique, not only because am I the researcher but also because I am a member of the culture being studied, the culture of ABI.

The methodological journey
After much reading and reflection leading to seemingly interminable incubation (Wallas, 1926), I wrote my MA thesis as an autoethnography. Although I was in a quandary at the time about the process of finding the most fitting method, in retrospect it was an educational and enlightening exercise. As Denzin and Lincoln (2000) have suggested, I began by asking myself the question, How could I best gain knowledge about the world of ABI? The dilemma of finding the optimal epistemological place in which to situate myself (Pinsent Johnson, Smith, & Thompson, 2003) when conducting the study haunted me as I devoured literature on the methods associated with the participatory paradigm (Denzin & Lincoln, 2000). What was my place in the study? Because of my unusual closeness to the subject I was researching, I knew that I was more than just the researcher. How could I most effectively tailor the fit of method to that of my role in the study?
I had originally chosen to use a personal experience narrative to explore the research question. This method is defined by Creswell (2002) as "a narrative study of an individual's personal experience found in single or multiple episodes" (p. 524). However, as time went on, I found that I needed to narrow the method down even more. Sustaining an ABI had not only redirected my life; the experience had changed the person I was into the person I am now. Irrevocably, I am now different. I was afraid that my experiences would color and influence my interpretations of the participants' experiences. Although I was very aware of the possibility that this might happen, I felt that just recognizing the danger of this occurring would not be enough. I had to take action and adopt a position that would not only legitimize my voice but also allow me to express my thoughts without marginalizing the voices of the participants. I knew that I had to be careful when choosing the method so that I would be able to convey the results of this study optimally.
It became clear that because I was examining the culture of a group, in this case people who had sustained ABIs, the method I would eventually use should incorporate ethnography. Ethnography has been defined by Chambers (2000) as an "inquiry that aim[s] to describe or interpret the place of [a] culture in human affairs…composed of those understandings and ways of understanding that are judged to be characteristic of a discernable group" (p. 852). I used the definition of culture proposed by Wolcott (1999): "Culture is an account of particular social processes as practiced by particular people in particular settings" (p. 253). This study describes and interprets the culture of ABI, a culture that has distinguishing qualities, as does any culture. Some of these qualities make aspects of reintegration into society difficult after an ABI, for example, when survivors try to fit into old social roles. Consequently, individuals who have sustained ABIs sometimes feel like outsiders in society (Seaton, 1998).
Because of the unique qualities of the culture of ABI, ethnography appeared to be an appropriate method. I questioned, however, whether ethnography would give me enough latitude to address fully the study's purpose, which was to explore the use of creative activities to enhance the selfesteem of individuals who have sustained an ABI. I felt I needed to include my opinions, views, and feelings on the subject, because I am from the same culture as the study's participants. My experiences mattered too, and my input, in terms of reflections, feelings, insights, and experiences, appeared to me to be integral to the study. However, if I used the traditional method of ethnography, whereby I would be in the standard role of researcher, I would be forced to ignore my experiences, which were guaranteed to produce very relevant data. I realized that my input might be invaluable in the context of this particular study. My contributions, as the researcher, would be as valid as those of the participants. I was as much a part of the ABI culture as they were.

The decision: Autoethnography
As Jenks (2002) has neatly summarized, my experiences undoubtedly have affected what I observed, what I wrote, and how others will interpret and react to what I wrote. Therefore, in the end, it was inescapable that I chose to use autoethnography as my method. By definition, autoethnography enabled me to tell the story of my life-changing experiences of ABI and incorporate my views, thoughts, and story to enrich the ethnography of my participants (Denzin, 1989;Reed-Danahay, 1997). Using autoethnography permitted my experiences to play a valid role in the study, because the genre includes the researcher as a participant. As Gergen and Gergen (2002) eloquently stated, "In using oneself as an ethnographic exemplar, the researcher is freed from the traditional conventions of writing. One's unique voicings-complete with colloquialisms, reverberations from multiple relationships, and emotional expressiveness-is honored" (p. 14). Therefore, autoethnography allowed me, in my role as the researcher who has sustained an ABI, to add my views and thoughts of the experience to enrich the story for the readers (Smith, 2004). Fittingly, Creswell (2002) has listed autoethnography as a type of narrative research, so, according to Creswell, I was still using my original choice of method.
The genre autoethnography enhances the study for the readers by allowing me, in my role as researcher, to inject my interpretations of my own experiences. Because I am from the same culture, I can, as Goodall (2000) has suggested, look at my own story through the same lens that I am using to interpret the worlds of my participants. In this study, although I was a researcherparticipant, I most certainly did not passively record and report. I invariably found I had to relinquish my views and understandings to grasp the full significance of what the participants were saying (Marshall & Rossman, 1999). I was able to emphasize the importance of the recovery process (Wolcott, 1999) for all the participants, giving them all equal voice. All of the participants realized that their participation in the study could potentially benefit others. They could demonstrate the positive influence of using creative activities as part of the process of ABI rehabilitation. They felt important and, most of all, respected. I made it clear to them that by doing creative work as part of their rehabilitation, they were helping to establish the significance of this type of rehabilitation for other survivors of ABI. Because the study was an autoethnography, my reality, as researcher-participant, is seen through a unique window to the world (Mykhalovskiy, 1997), one that could have been experienced only when participating actively in the study.

What is autoethnography?
By using autoethnography, researchers can use their experiences, together with those of other participants, to complement their research. Autoethnographies can use alternate forms of representation, such as short stories, poems, and artistic interpretations. The term was originally coined by Hayano (1979) to refer to anthropological studies by individuals of their own culture. The exact definition of the term is elusive, and there are many other genres, too numerous to list, that fall under its umbrella (Ellis & Bochner, 2000). Behar (1996) has described emerging genres, such as autoethnography, as efforts "to map an intermediate space we can't quite define yet, a borderland between passion and intellect, analysis and subjectivity, ethnography and autobiography, art and life" (p. 174).

Being the researcher
As a participant in this study, I, too, had to complete a creative activity. I did not know what to do. I could paint with acrylics, or I could paint pottery, things I do on a regular basis. It finally occurred to me that completing this thesis would do wonders for my self-esteem. My creative project could be my master's thesis. I had spent a long time in recovery and then more time completing the MA course requirements before embarking on the thesis. It had been 7 years since the accident, 7 years since I had completed a major project. I was satisfying the guidelines I had set for my participants: The thesis was a creative project. Because I am a creative person, I had already intended to take full advantage of the alternative representations of the paradigm I was using. In the end, I included a short story of my personal journey and a play written from a focus group interview, and I painted a 4´ x 6´ (1.2 x 1.8 m) pictorial representation of the data analysis (Wolcott, 1994).
These elements contributed richness to the data, both because they are innovative and because they allowed me, as the researcher, to provide unique glimpses both of my own story and of the study. The short story added my interpretation to the long recovery I had undergone. I used the metaphor of a long tunnel for my recovery. I tried to write the story as I was during specific times. For example, initially I was speaking in short sentences and walking very unsteadily. The unsteady walking is symbolized by deep, shifting sand on the tunnel floor. The focus group that I wrote as a play lets the reader peek into the unique relationships I had with the participants. I was able to show that the participants had no trouble teasing me; this indicates a level playing field between us. They were open and frank among themselves but also when they were one on one with me. The painting of the data analysis showed the participants, portrayed as fish, swimming among the themes, symbolized by seaweed. As well as being a very effective visual representation, portraying the analysis in this way let me use my creativity.

The participants
When researchers are becoming familiar with a new method, it is incumbent on them not only to recognize the positives that have drawn them to the methodology but also to make themselves aware of its potential shortcomings. There is a fine balance between the added insight that autoethnography allows and the lack of clarity that can occur when researchers do not define their positions clearly within their autoethnographical studies. One of the potential dangers that exist is that in their drive to become as close to the participants as possible, the fine line that researchers draw between themselves and their participants will be too blurry. For example, in an effort to become privy to more information from Sally (all names are pseudonyms), one of my participants, I told her too much about myself. As a result, she "crowded my space" and tried to put herself in a friend role (Smith, 2004). The questioning and probing of my personal experience that occurred made me extremely uncomfortable. Unbeknownst to me, I was experiencing transference, which occurs when clients identify too closely with therapists. Sally was trying to "fast-forward" our relationship-what she would know about me, what I would tell her-without my consent. I was not prepared to change the participant-researcher relationship we had into the relationship she wanted, one in which we would become confidantes and friends. As soon as I left Sally's apartment that day, I called my therapist, who explained what she suspected had happened. This incident caused me to realize that in the future, I would have to examine carefully how to tailor my relationship uniquely with individual participants to maximize their contributions to the study.
Another experience of refining the researcher-participant relationship to optimize the data I gathered was more successful. Jim was bright, cheerful, and eager to contribute. A stroke had left his speech very difficult to understand, so rather than recording conversations, he would e-mail me comments and thoughts regularly. These e-mails were observant and full of rich information. I found, though, that I needed to refine our relationship to glean yet more insight into his character, so I suggested that we join a hockey pool together. Although this was something that was completely separate from the creative activity of the study, it could not have been a better move! I was inundated with e-mail explaining what was happening (I know nothing about hockey) and what our current position in the pool was. The serendipitous move was fortunate. I gained much insight into the kind, generous man who was my most forthcoming participant. This showed me, as Sally had, how I had to consider each participant individually. My relationship with each one would be different.
There was one unanticipated and very positive result from this study. All participants felt that this was "their" study. This factor added hugely to their self-esteem. Jim was no end of help; he emailed me faithfully and answered all questions with careful consideration. Diana made a commitment to meet every week; she was always there, full of meaningful conversation. Sally penciled in our weekly time; she was attentive and always sent me e-mails of her thoughts and reflections. After the focus group interview, I invited the participants to lunch. Only Diana and Sally could come, but they talked nonstop, to the point where I finally had to go, and so I left them, still talking! Working with Valerie gave me insight into how to interact with individuals who are significantly cognitively impaired. Valerie's caregiver made sure she was always there for my meetings with Valerie.

Conclusion
Because I am also an ABI survivor, using autoethnography enabled me to explore my subjective and cultural experiences as well as those of my participants. Autoethnography allowed my personal experiences to become valid data. I was able to research, explore, and use a relatively new genre for the purposes for which it was intended. Autoethnography freed me to write reflectively, thoughtfully, and introspectively about a very personal subject close to my heart. Completing this autoethnography contributed to my healing, because, as Behar (2003) commented with reference to her experiences, it helped me to come "a full circle, or better, the circle widened, stretched, opened for me" (p. 342).
I believe I enriched and added credibility to the research and exploration of a marginalized and very specialized population by using autoethnography. None of the participants chose to read the study, but they were pleased to hear that it had been nominated for a prize at my university. As a result, my first foray into qualitative research has, I hope, opened many eyes to the realities of life after an acquired brain injury and to the possibilities that autoethnography offers as a method. As a method for someone with a unique perspective, or from an "other" culture with a one-of-a-kind story to tell, autoethnography affords the opportunity for self-reflection. At the same time, it gives voice to others from the researcher's culture.


Aging well: toward a way of life for all people.

The world's population as a whole is living longer. Population aging is a trend that will bring new social, political, and economic demands to countries throughout the world (1). Without proactive steps to create individual and environmental changes that promote aging well, the escalating costs of health and social support systems for an aging population will be unsustainable both domestically and worldwide (2).

The concept of aging well, which is based on a nonmedical approach to promoting health and well-being, is fundamental to increasing length and quality of life. Aging well promotes personal behaviors and life-course environments that limit functional declines, especially those caused by chronic conditions, to help older adults maintain their independence and health. Aging well emphasizes the idea that people can maintain satisfying and healthy lives as they age by exercising the choices that optimize healthy, active, and secure lives (1,3,4). Aging well is a dynamic, interactive process that creates long-term, positive change by involving individuals in the physical, social, economic, historical, and cultural contexts of their environments.
Development of an aging-well framework fits well with articulated health goals for the United States and the international community (5). While many countries have made significant gains in addressing national health objectives, a nonmedical framework for aging well has not yet been fully developed. This knowledge gap persists in spite of evidence linking one or more specific areas, such as social support, physical activity, and material security, to longevity and quality of life.
Although no single formula automatically results in aging well, research conducted by Fry and colleagues provides several illustrations of factors that contribute to aging well in different cultures (6). For example, in Hong Kong a comfortable old age is marked by family and social qualities that demonstrate open-mindedness and tolerance. In the United States, engagement, vitality, and activity are associated with aging well. As Fry and colleagues point out, "Having good health and comfortable pensions does not hurt either." Across the cultures studied, determinants that are important to aging well -including physical health, mental effectiveness, material security, social resources and relationships, and meaningful daily activity -are common, but their relative contributions may vary.
Research supports the concept of aging well (1,5,(7)(8)(9)(10)(11). Current international research efforts are moving forward from identifying the determinants of aging well to testing an integrated model and investigating viable individual and environmental indices of aging well. These research initiatives will help to identify populations, environments, services, programs, and public policies that support the continued involvement of older adults in positive, productive, and healthy living.  The focus of GARNet's work is a collaborative project to examine the social and environmental determinants of aging well. This project is using comparable data sets from the 12 member nations to develop an individual index of aging that is comparable within and across countries. This index will give policy makers a clearer and more comprehensive understanding of the factors that affect aging well. These findings can be applied in developing and evaluating social policies and services that will promote aging well across the life span.
The GARNet index measures items in each of the following five domains of aging well: physical health and functional status; mental/cognitive effectiveness; social support resources; daily life activity; and material security. A broad range of research and theoretical literature supports the importance of each domain.

Physical health and functional status.
In cross-cultural studies of aging, good physical health and functional status have been associated with aging well (6). According to studies conducted by the MacArthur Foundation Network on Successful Aging, poor health in old age is largely attributable to the effects of multiple lifestyle choices and life conditions, such as physical inactivity, poor diet, and smoking, that increase the risk for disease or disability (11). Thus, maximizing physical health involves reducing these risk factors for disease and disability (11). Indicators for this determinant include health status, doctor and hospital visits, medication and supplement use, assistive device use, health promotion behavior, health risk behavior, and measures of functional independence associated with activities of daily living.
Mental/cognitive effectiveness. A critical domain of aging well is cognitive functioning, especially the cognitive effectiveness necessary to maintain the daily activities associated with overall health. Research by Baltes described two fundamentally different influences on the aging mind (8). The first, cognitive mechanics, is genetically endowed; the other, cognitive pragmatics, is culturally learned. Baltes speculates that older adults who lead effective lives use a framework of self-management and practical knowledge (cognitive pragmatics) to optimize gains and compensate for losses in cognitive mechanics. This view of cognitive efficacy fits well with descriptions of aging well. Indicators in this domain include measures of self-esteem, perceived control, resilience, and mental well-being.
Social support resources. A strong network of friends and family has long been recognized as an important contributor to good health in old age. Isolation or the lack of a strong social network places people at increased risk for poor health. Furthermore, a strong social support network can actually buffer or reduce some of the effects of aging (11). Indicators in this domain include measures of the availability and use of social support, network types, and other social resources.
Daily life activity. Identity in old age is influenced by the roles assumed in society. Because adult life involves cyclic and complex stages of personal development mediated by transitions, role disturbance and continuity are significant determinants of health in later life. For example, retirement potentially removes a lifelong source of identity and may create multiple points of disturbance to a person's well-being. An old age of idleness, without purposeful activity to help maintain identity and health, may well undermine aging well. The MacArthur Foundation studies in successful aging examined productive activity in a broad sense, including "any activity, paid or unpaid, that generates goods or services of economic value" (11). This approach allowed the researchers to examine a wide range of roles through which older adults can contribute to society. They concluded that three factors have a major influence on productive activity in older adults: 1) health and functional capacity, 2) social support network, and 3) personal characteristics. This domain includes indicators of engagement in formal and leisure activities, as well as other measures associated with daily time use.
Material security. Another important domain of aging well is material security (6). Having the material resources necessary to secure housing, sustenance, and access to needed services (e.g., medical care, social support) is fundamental to health and well-being in later life. In devel-oped countries, retirement practices complicate the assurance of health and well-being in old age. For most of the past century, an ideal associated with retirement policies was to provide for the financial security of older Americans through social security and pensions; however, many older people, including women, minorities, unemployed people, and lower paid workers have not realized that ideal (6). Material security remains precarious for some older adults, such as very old people who have exhausted all of their resources. All individuals make decisions on the basis of providing for their basic needs (e.g., food, housing, safety) and for the basic needs of their families. When basic needs are unmet or compromised, personal health is compromised. The relationship between material security (conceived in broader terms than just income) and health is fundamental to aging well. The indicators in this domain include measures of access to shelter, food, services, safety, and economic resources, which are essential for health promotion and protection in all cultures worldwide.
These five domains are each individually well documented in the research literature. What remains to be determined, however, is how they operate interactively to circumscribe an individual's success at aging well. This is the question that GARNet's research is seeking to answer. By providing a more robust and universal understanding of the aging experience that goes beyond the medical perspective, the answer to this question could help us build a stronger foundation for health promotion and disease prevention initiatives.
Public health research and policy initiatives should embrace new paradigms that move health beyond the medical model toward a larger, more encompassing framework. The generation and testing of innovative frameworks that provide explanations of aging well are imperative for containing and reversing the impact of chronic disease and disability in old age. The aging of the population will only intensify this imperative. GARNet's findings, as well as those of others working in this area, are intended to lead us closer to a world where age is celebrated and aging well is a possibility for all older people.


extfieldsofstudy: Biology Medicine

text:
Biology by Numbers—Introducing Quantitation into Life Science Education

An online educational module introduces students to concepts of quantitation and numerical simulations in developmental biology

D riven by the massive datasets that are generated by "omics" research, the molecular life sciences are entering a new phase. This phase is characterised by a shift in focus from individual genes and their products to networks and whole systems [1,2,3]. For a thorough analysis of the behaviour of networks and their underlying principles, quantitative tools are often necessary. Numerical simulations can, for example, be used to explore the behaviour of a network when the values of different parameters are varied, and, in turn, mathematical analysis can help to understand a particular biological phenomenon [2].
The successful application of quantitative tools in the molecular life sciences requires a good understanding of these tools and suffi cient knowledge of the biological system under study. This can be achieved by collaboration between quantitatively trained scientists such as physicists on the one hand and biologists on the other. However, cultural differences hamper such collaboration [1]: even at the undergraduate level, students in the different disciplines speak very different languages [4].
A more productive approach is therefore to prepare students better for the quantitative nature of the molecular life sciences by integrating quantitative thinking and biology in the life science curriculum. This can be achieved in various ways. For example, a curriculum could be developed in which mathematics, the physical sciences, and biology are introduced together [4]. However, we recommend that quantitative thinking also be included throughout the curriculum in the biology courses themselves, covering topics such as cell biology, developmental biology, and biochemistry. We consider this important because it will help to show students how quantitative tools can be used to address various cutting edge questions in biology.

A Modelling Module in Developmental Biology
As an example of the integration of quantitative teaching and cutting edge biology, we have implemented an educational module in which numerical simulations are used in an existing course on developmental biology (http:⁄⁄mbedu.fbt.eitn.wau.nl/ demo_plos/). Some of the features of this module and the thinking that led to its development are quite general, and so we present the module here as a case study in the hope that this might inspire and guide others to create similar resources.
First, we wanted to illustrate to students the value of using numerical simulations to study a developmental process. Therefore, a pattern-forming mechanism was selected that can initially be rather hard to understand: the generation of the morphogen gradient formed by the extracellular signalling molecule decapentaplegic (Dpp) early during Drosophila embryogenesis [5]. The generation of this gradient results from the fact that

Figure 1. A Simulation That Students Can Perform
After several minutes, Dpp forms one peak in the centre of the dorsal region, as in the wild type. The various elements of the quantitative model can be entered under "protein conc. changes", "initial localizations", "values of constants", and "initial concentrations". The numerical simulation itself shows the dynamic behaviour of the designed quantitative model. key proteins are synthesized in different embryonic regions, from the formation of complexes of these proteins, and from the different diffusion rates of these complexes and their components, as well as from the specifi c degradation of some components. Students are guided through the creation of a model for Dpp gradient formation based on a set of experimental data. At several stages, students can perform simulations in a separate simulation environment. Students use simulations, for example, to check whether a number of core interactions is suffi cient to yield the most important characteristics of the wild-type gradient.
Second, we designed the simulation environment in such a way that biology students with their existing mathematical background can build quantitative models and run numerical simulations themselves. In this environment (Figure 1), students do not have to program anything, or set up differential equations, themselves. Instead, they indicate which processes occur at the molecular level, and the program then shows how each of these processes is translated into a term in a differential equation. In Figure 2, for example, the program adds a diffusion term to a differential equation if the student indicates that diffusion occurs.
Besides setting up the equations in this way, students specify the initial localisations and concentrations of the different proteins, as well as the constants that are used in the differential equations.
Third, we wanted to make sure that students would use the simulation environment effectively. Therefore, a clear goal is formulated when students enter the simulation environment. For example, they are asked to make a model that generates a Dpp gradient that fulfi lls a number of specifi c criteria, or simulates certain mutants. After running a simulation, students can view feedback that helps them draw conclusions or consider the next step to be taken. If a student's model, for example, generates a gradient that is too shallow, the student has to indicate which change in the model he expects to be useful for generating a steeper gradient. The student then receives an intuitive explanation of the usefulness of the given suggestion. If an increase in the synthesis of one of the proteins, Short gastrulation, is proposed, for example, feedback is given that this could indeed be useful, since there would then be more Short gastrulation available to transport Dpp, such that the gradient can become steeper. In this way, the student is stimulated to carefully consider each step and is provided with suffi cient support to decide which is a useful step to follow. In addition, with this type of feedback, explanations are given that relate quantitative changes in the model to qualitative changes in its behaviour, which should increase the student's understanding of the behaviour of the biological model.
We consider it important that students, while using the module, are not distracted too much by quantitative issues from the actual biological principles and facts. These have to be mastered in order to obtain a strong biological background. If students want to learn more advanced quantitative skills, they can still follow courses that are specifi cally aimed at this aspect.

The Future
Quantitative analysis is already gaining importance in molecular life sciences. Therefore, it is desirable that curriculum changes are implemented in the short term. This poses challenges to faculties, especially to those whose members do not have much, if any, experience with the application of quantitative tools in their own research. Therefore, it may be useful to initially focus on the development of learning materials that are rather selfcontained, such that their application requires relatively little competence in quantitative analysis from the teaching staff. If these materials are openly available they can be incorporated rapidly into existing courses, such that even the current generation of students may be better prepared to integrate quantitative thinking and biology in their future research.


extfieldsofstudy: Mathematics Medicine Biology Physics

text:
Power-law scaling in dimension-to-biomass relationship of fish schools

Motivated by the finding that there is some biological universality in the relationship between school geometry and school biomass of various pelagic fishes in various conditions, I here establish a scaling law for school dimensions: the school diameter increases as a power-law function of school biomass. The power-law exponent is extracted through the data collapse, and is close to 3/5. This value of the exponent implies that the mean packing density decreases as the school biomass increases, and the packing structure displays a mass-fractal dimension of 5/3. By exploiting an analogy between school geometry and polymer chain statistics, I examine the behavioral algorithm governing the swollen conformation of large-sized schools of pelagics, and I explain the value of the exponent.

Brief history of studies on the packing geometry of fish schools
Animal packing in social aggregations is of fundamental interest in ecology, and their conformations have been extensively studied (Parr, 1927;Symons, 1971;Okubo and Chiang, 1974;Graves, 1976;Pitcher and Partridge, 1979;Aoki, 1980;Partridge, 1980Partridge, , 1982Dill et al., 1981;Aoki and Inagaki, 1988;Misund, 1993a;Parrish and Hamner (Eds.), 1997). Pioneer tank observations of pelagic fishes (herring, sprat and mackerel) were conducted by Parr (1927), and the "persistently uniform density" of a school was noted. From experiments on schools of saithe Pollachius virens, herring Clupea harengus and cod Gadus morhua cruising in a ten meters circular gantry tank, Pitcher and Partridge (1979) validated that all the fish in a school occupy a volume proportional to Nb 3 , where N is the number of fish and b is the mean nearest-neighbor distance (approximate to the mean fish-body length) in the school. Misund (1993a) reported from field observations that the number densities of herring schools are almost independent of the dimensional size of school but are an order of magnitude lower than the packing densities observed when schooling in small tanks.
Tauti and colleagues (Tauti and Miyosi, 1929;Tauti and Hudino, 1929;Tauti and Yasuda, 1929, 1930, 1933a had experimentally shown that a fish school can be viewed as a group of inorganic particles and treated with the methods of physics. Breder (1954) explicitly applied physical equations to such fish schools. Recently, a number of theoretical and numerical models of schooling and flocking behavior have been studied by biologists, mathematicians and physicists (Okubo and Levin, 2001;Camazine et al., 2001;Vicsek (Ed.), 2001). As for the animal group geometry, many models for social aggregations, however, predict that densities increase as the group size (in number) increases [overviews of such models are given in Refs. (Warburton and Lazarus, 1991;Beecham and Farnsworth, 1999)]. Mogilner et al. (2003) mathematically tackled this problem of constant density and revealed the condition for a well-spaced group, i.e. what class of mutual interactions results in a relatively constant individual distance in the interior of the aggregate.
More recently, by means of underwater acoustics, the school sizes (in number or biomass of fish) have been quantitatively measured for different values of the dimensional size of schools in the wild. Precise data on conformations of large-sized schools of pelegic fishes became available rapidly, which were extremely helpful in elucidating a certain geometric law, i.e. the relation between dimensional and biomass sizes of pelagic fish schools, bringing about some important changes in our viewpoints. Misund and colleagues (Misund, 1990;Misund et al., 1992;Misund, 1993b;Misund et al., 1995Misund et al., , 1996Coetzee, 2000;Misund and Coetzee, 2000;Misund et al., 2003) found that the powerlaw scaling in dimension-to-biomass relationship exists, and is robust across a broad range of pelagic species as well as across diverse environments. They demonstrated that if the biomass N in a school is, say, doubled, the crosssectional area of a school is increased by a factor 2 2ν , i.e. cross-sectional area ∝ (biomass) 2ν , and that the exponent ν looks universal, reading 0.5 (from the field data ν ranges from 0.415 to 0.77). The geometric law they found implies that the mean density of a school scales as N 1−3ν in three dimensions of space and the conformation of social aggregations swells (i.e. ν > 1/3). Such a relationship as Eq.
(1) with ν = 0.5 has been utilized for the dimensions to biomass conversion (e.g. transforming the school diameter R to the number N of fish) in analyzing school-size distributions (Anderson, 1981;Niwa, 1996bNiwa, , 1998Niwa, , 2003Niwa, , 2004a. It might come as a surprise that packing densities decrease as the group size increases, contrary to previous observations and predictions. Laboratory observations for school geometry and internal structure have been made exclusively in small tanks, which generally show the constant density of fish that the school volume is proportional to the number of individuals. In all set-ups, the factors resulting in homogeneous, cohesive school may be especially pronounced, and it therefore seems that the quantified structure is skewed. Pitcher and Parrish (1993) claimed that homogeneity in schools has been over-emphasized. In-situ observations of herring and sprat schools with a high resolution sonar revealed that the packing structure within the schools is rather heterogeneous (Cushing, 1977). This has been confirmed by measurements of free-swimming schools using photography and high-resolution echo integration, which showed that the packing density distribution in capelin Mallotus villosus and clupeoid schools varies considerably (Fréon et al., 1992;Misund, 1993a). Regions of high density are usually found within the schools, and even empty vacuoles have been recorded. Misund and Floen (1993) observed by repeated echo integration that there were large variations in internal packing density of herring schools (i.e. high-density regions or empty lacunas within a school), and that the packing density structure was quasi-stationary. Besides artificial environments in small tanks, the discrepancy in former observations could have been caused by too small numbers of fish in the schools. For instance, in Pitcher and Partridge (1979), N takes a few tens of fish. Since such a geometric law above is always defined only in a certain limit (de Gennes, 1979), the scaling in the dimensionto-biomass relationship is expected to hold for large-sized schools of pelagics.
In this paper, the exponent ν is estimated according to the established universal scaling law in the school-size distribution of pelagic fishes (Niwa, 2003(Niwa, , 2004a: choose the suitable value of ν to achieve the best data-collapse on the size distributions in terms of the school dimension. Notice that the dimensionto-biomass relationship is a property of the single school and the scaling exponent ν is determined by the behavioral algorithm of fish schooling at individual level, while the scaling in the school-size distribution emerges from the interschool interactions at population level (i.e. a global property of the interacting school system).  Hara (1990) off southeastern Hokkaido in the summer, 1982. The same data are available in Hara (1984). Small schools of vertical dimensions in the first bin (< two meters) may be stray ones.

Scaling in school-size distributions
Animal group size is a focal issue in ecology that, in contrast to scaling, has introduced a single preferred size (i.e. optimal or compromise size) for any organism living in groups (Pulliam and Caraco, 1984;Higashi and Yamamura, 1993;Niwa, 1996b;Hoare et al., 2004). Figure 1 shows an example of the histogram of school dimensions of Japanese sardine Sardinops melanostictus. A peak frequency and a right skew are typical of pelagic fishes (Anderson, 1981;Niwa, 1996bNiwa, , 1998. Their linear dimension, e.g. the vertical thickness, of the school falls into a certain range below a few tens of meters. One fish may not be the right atomic unit in schooling, since field observations suggest that no school exists under a certain minimal size. This may cause binned data of school sizes to exhibit the fake peak frequency. Anderson (1981) and Niwa (1996b) have ignored slowly decaying (or fat-tailed) distributions, including scaling laws, in such data of school sizes short ranged with a fake peak frequency (the possibility of power laws was already presented in their models but not exploited). A traditional, widely used Gauss statistics says that, for the data from Hara (1990) shown in Fig.1, finding sardine schools ranging from 18 to 20 meters in vertical thickness should only occur about once every 10 9 detections of schools [for detail, consult Niwa (2004a)]. In other words, it is not the real world! Aquatic observations actually say that finding such schools occurs about once every 500 detections. The probability that such schools are found is 10 6 times as large!
The possibility of scaling in such distributions was found by Bonabeau and Dagorn (1995). Fat-tailed distributions have been found to quite generally character-ize the size heterogeneity of pelagic fish schools (Niwa, 1998;Bonabeau et al., 1998Bonabeau et al., , 1999. Only lately, Niwa (2003Niwa ( , 2004a showed that the school-size distribution of pelagic fishes has the universal functional form of a power-law decay and a crossover towards the exponential distribution.

Scaling and data collapse in ecology
Scaling laws have been found to characterize many patterns in ecological systems (Azovsky, 2000;Chave and Levin, 2003;Halley et al., 2004). There is an empirical rule about the relationship between the areas (L) of island and the numbers (S) of species on them: whether counting birds, butterflies, plants or copepods in ponds, a tenfold smaller area contains roughly half the species. This can be fitted by a power-law function with the exponent z ≈ 0.25 (Preston, 1962;MacArthur and Wilson, 1967;Harte et al., 1999;May, 1999). The species-area relationship is widely used in rough assessments of likely future rates of species extinction (because observed rates of tropical deforestation are equated to loss of habitat area, we can assess the annual production of species committed to extinction). So, a more secure understanding of such relationships has great practical importance (May, 1999). Banavar et al. (1999) applied finite-size scaling (Fisher and Barber, 1972;Binder and Heermann, 1988) to test the hypothesis of scaling invariance in the species-area relationship resting on a model proposed by Harte et al. (1999) for the species abundance distribution. Following the conventional notation in the physics literature, the system sizes are hereafter denoted by L.
The finite-size scaling (FSS) hypothesis assumes that the fraction of objects with size n for a finite system of size L is written, with a scaling exponent A, as P (n; where F is a universal scaling function. P (n; L) reads, for instance, the speciesabundance distribution, which defines the probability that any given species on a census patch of area L has n individuals, where P (n; L)S(L) is the expected number of species with n individuals. L A is a measure of the width of the probability distribution P (n; L), e.g. the mean or the standard deviation of the distribution. Equation (3) expresses the principle that the behavior of the finite system, i.e. P (n; L), is controlled by the ratio n/L A . We test the FSS hypothesis by data collapsing: when y = L A P (n; L) is plotted versus x = n/L A , if the distributions across different L's fall on a single curve (e.g. collapsing distinct abundance distributions measured in different areas and with different total numbers of individuals onto one scaling curve), then we should be able to identify a scaling function F (x). In order to determine the exponent A, we try a best-fitting procedure such that the family of curves P (n; L) collapses onto a single curve as well as possible. We then obtain other scaling exponents (e.g. z of the species-area relationship) resting on the scaling relation between scaling exponents (Goldenfeld, 1992). By noting that the total number of individuals of all species is equal to n nP (n; L)S(L), the species-area relationship leads to a scaling relation assuming that the total number of individuals of all species is proportional to area L (Banavar et al., 1999).
The universal scaling is fundamental to data collapse. The power-law scaling emerges as ubiquitous properties in ecology. In statistical analysis in ecology, data collapsing across different environments or species has been, however, observed only lately in dynamics of breeding bird populations (Keitt and Stanley, 1998;Keitt et al., 2002), food web structure (Camacho et al., 2002), microbial body-mass spectra in marine ecosystems (Rinaldo et al., 2002), school-size distributions of pelagic fishes (Niwa, 2003(Niwa, , 2004a, and in the context of ecological economics, capture fisheries productions of countries (Niwa, 2004b).
2 Scaling in dimension-to-biomass relationship

Estimation of the exponent ν through data collapse
In order to understand the geometric properties of school configurations, the school biomass N must be measured for different values of dimensional size R, and we must compare them. Misund and colleagues extensively performed the simultaneous observations of the values of R and N for pelagics by the following two methods: (i) two-dimensional (cross-sectional) acoustic measurements (unit in square meters) and subsequent purse seine capture (unit in tonnes) of schools (Misund, 1990(Misund, , 1993b, and (ii) acoustic measurements of the threedimensional structures and backscattered echo energy (Misund et al., 1992(Misund et al., , 1995(Misund et al., , 1996Coetzee, 2000;Misund and Coetzee, 2000;Misund et al., 2003). The surveys were conducted on stocks of anchovy Engraulis capensis, herring Clupea harengus, horse mackerel Trachurus trachurus capensis, mackerel Scomber scombrus, pilchard Sardinops ocellatus, round herring Etrumeus whiteheadi, saithe Pollachius virens, sardine Sardinops sagax, and sprat Sprattus sprattus, in different seasons and geographic regions (the Barents Sea, the North Sea, the Norwegian Sea, the northeastern Atlantic and off Namibia, and off the coast of South Africa). They found that there is some biological universality in the dimensions-to-biomass relationships, Eq.(1). Their finding implies that the radius R of the school scales as in a statistical sense. N, denoting the school biomass, is hereafter defined by the number of fish in a school.
Now I propose another way to establish the scaling law for school dimensions, Eq.(5), by applying universal scaling law in fish school-biomass distributions (Niwa, 2003(Niwa, , 2004a. The biomass distributions W (N) follow a power law with exponent β = 1 up to a cut-off size N P , where P (x) is a crossover scaling function with a strong drop for x > 1, and the cut-off size (crossover size from power-law to exponential decay) is calculated from the biomass histogram data {(N i , W i )|i = 1, 2, . . .}, where histogram bins are chosen with width ∆N.
The dimension data of fish schools are binned with width ∆R, giving the set of frequencies (5) and (6), the distribution of geometric dimensions of fish schools is represented as where and Therefore, the school-dimension distribution follows a power law with the same exponent "−1" as the school-biomass distribution. The power-law distribution W (G) (R) is truncated at a cut-off size R P , which is also calculated from histogram data of school geometric dimensions, The following normalizations are adopted for the scaling function P (G) (x) and the histogram data of geometric dimensions of fish schools, because the cutoff size N P is proportional to the total number of fish in the school system [Niwa (2003[Niwa ( , 2004a; see also Eq.(23)]: respectively.
Fat-tailed school-size distributions are necessarily truncated because the population is finite. Since the size R P depends on the exponent ν following Eq. (11), so that we can determine the value of ν by evaluating the location of the cut-off in the power-law distribution W (G) (R). The ordinary least squares regressions might, however, lead to a "wrong" estimation of the exponent (Niwa, 1998;Bonabeau et al., 1998Bonabeau et al., , 1999. I make use of the data collapse to extract the "right" exponent. From Eqs. (8) and (12), when y = W (G) R P is plotted against x = R/ R P with correct parameter ν, all the empirical data should collapse onto each other. The power-law exponent of school-dimension distributions, ν, is then evaluated through data collapse. Let us search for the value of ν that places all the points most accurately on a single curve. We use a set of histogram data of vertical dimension of Japanese sardine Sardinops melanostictus schools, from 22 acoustic surveys by Hara (1990) off southeastern Hokkaido for seven years, 1981years, -1987years, . Hara (1986 reported that Japanese sardine migrate as a huge-sized school in number from a few hundreds of thousands to a few million of fish. To obtain the best data collapse, the x-axis is divided into bins (Fig.2a), and for each bin two-dimensional variance is calculated, where σ x and σ y denote the standard deviation of the mean µ x and µ y , respectively. The parameter ν is then estimated at value that minimize the mean of two-dimensional variance for the bins (Fig.2b). The mean of twodimensional variance, ǫ, is a measure to determine the goodness of collapse (Lillo et al., 2002(Lillo et al., , 2003Niwa, 2004a). A good data collapse can be obtained by using the value ν ≈ 0.6. The resulting plot of empirical school data is shown in Fig.2a. Experimentally fitting the parameter ν to achieve a good data collapse, "3/5", is reminiscent of the Flory value of the exponent in a powerlaw dependence of the coil radius of a polymer chain (in three-dimensional solutions) on the degree of polymerization (Flory, 1953;de Gennes, 1979).
Here we see that the power-law regime of the distribution is too short, which misled Anderson (1981) and Niwa (1996b) into overlooking the power-law distributions of school sizes. Notice that many power laws that appeared in the ecology literature span less than two orders of magnitude of scale (Halley et al., 2004). The power law range of too few scales is not unique . . , 6. The rectangle in gray reads the interval µ y ± σ y , i.e. the error σ y on the mean µ y (indicated by the slit) for each bin. The solid line is a prediction of the mean-field theory (Niwa, 2003). (b) The mean of two-dimensional variance, ǫ, versus the power-law exponent ν. Although ǫ shows noisy fluctuations, it takes a minimum around ν = 0.6. Data from Hara (1990) are analyzed.
to ecology; the largest numbers of power laws reported in the physical science are for small ranges (Hamburger et al., 1996).

Retest of the FSS in school-biomass distributions
The acoustic-survey data are converted into a school-biomass histogram as follows We now crosscheck the value of ν through finite-size scaling (FSS) analysis of school-biomass distribution (Niwa, 2004a). Since the finite population size causes the truncation of power-law distribution W (N) ∝ N −β , there is a welldefined quantity which depends on the system population size. In order to characterize the finite size effects, FSS hypothesis is used: the distribution function depends on N only through the ratio N/L A , where F (x) is a universal function independent of fish population size. The prefactor L −B is required to ensure the normalization where P (N) [≡ N β W (N)] now represents the probability distribution of school-biomass sizes. From the FSS hypothesis, it is expected that when W L A+B is plotted against N/L A with correct parameters A and B all the data collapse onto a single curve. The power-law exponent of biomass distributions, β, is then evaluated through FSS analysis. The value of B/A is the estimate of the power-law exponent Let us analyze a set of 22 acoustic-survey data of sardine schools (same as Fig.2) converted into biomass histograms by using Eq.(15) with ν = 3/5. In a simultaneous best-fitting procedure (Fig.3), a good data collapse can be obtained by using the values A ≈ 1 and B ≈ 1. The power-law exponent derived from the FSS collapse is β ≈ 1. The resulting plot is shown in Fig.3a.
The school-biomass distribution follows a power-law decay with exponent −1, and is truncated at the cut-off size of Eq. (7). The FSS collapse confirms the scaling laws for school sizes, Eq.(5) with ν = 3/5 and Eq.(6) with β = 1.
Finite-size scaling techniques have been applied to ecology and shown to predict scaling relations between scaling exponents in relative abundance of species [Eq.(4)] (Banavar et al., 1999;Aji and Goldenfeld, 2001), in dynamics of breeding bird populations (Keitt et al., 2002), in biomass-size distributions of seston (Rinaldo et al., 2002), and in exploitation of fish stocks (Niwa, 2004b). In school-size statistics, we expect to have a scaling relation. We now choose the normalization where Φ denotes the total fish population in the school system ( i W i ∆N gives the total number of schools). Since Eq.(16) implies L A ∝ Φ γ with a scaling exponent γ, the FSS relation for the school-biomass distribution is written as Accordingly, the normalization of Eq.(20) yields the scaling relation The best-fitting value in data collapse, β ≈ 1, gives γ ≈ 1. As a consequence, the location of the cut-off in the power-law distribution of school sizes simply  (14)] are calculated for bins chosen equally spaced on a logarithmic scale as x ∈ 10 −1+(k−0.5)/3 , 10 −1+(k+0.5)/3 with k = 0, 1, . . . , 5. The rectangle in gray is same as Fig.2. The solid line is a prediction of the mean-field theory (Niwa, 2003). (b) The region of the AB-plane in which the minimum of the mean of two-dimensional variance exists. A measure of data collapse for scaling, i.e. the mean of two-dimensional variance, ǫ, takes a minimum ǫ min for the right choice of (A, B). The minimum is found with the precision i.e. width of the minimum, ∆ǫ = 10 −3 in black region (∆ǫ/ǫ min ≈ 3.12 × 10 −3 ). The values of the parameters lie in the intervals A = 0.999±0.008 and B = 1.003±0.024, and therefore β = 1.004 ± 0.032.

Gaussian model
Let us now investigate cohesive motion of schools in a three-dimensional space from the viewpoint of the behavioral algorithms which govern their formation and dynamics: attraction between neighbors maintains cohesion of the school; a tendency to align with neighbors produces collective motion of the school. The minimal model of cohesion is a linkage of neighbors consisting of harmonic spring, because the inter-fish distance follows a Gaussian distribution (Dill et al., 1981;Partridge, 1982;Niwa, 1996a). Using the relative coordinates to the center of school, the equation of motion of fish (as noisy self-propelled particles) in a large school of size N is written by a one-body approximation as the following Langevin equation (Niwa, 1994(Niwa, , 1996a: providing that individuals are sufficiently sensitive to behavior of their neighbors, where f (r) is the cohesive force; J denotes the strength of alignment; and η is a random perturbation of the velocity with strength ǫ v and a δ-correlated time dependence.

Self-consistent calculation of of cohesive force
The attractive interaction force acting on one body due to a system of bodies is the neighbor-joining harmonic spring. One essential approximation is to replace the many-fish problem by the problem of solving the motion of one fish in a certain self-consistent field. The cohesive force is then written in the following form by the one-body approximation: wherekN −α is the effective spring constant for the springs strung out from the center of the school to a fish: the number of consecutive springs joining the fish to the center of the school via other companions is proportional to N α . The total "elastic energy" of inter-fish bonds in the school depends linearly on number N of individuals in the school. The overall elastic energy after integration over a sphere of radius R (denoting the average radius of the school) results in E el ∝ R 4 N −α , which is derived from the ideal harmonic spring, Eq.(25). Therefore, the exponent α will be determined self-consistently, The solution of the Langevin equation (24) with Eq.(25) takes the following asymptotic forms (Hori, 1977): where the root-mean-square r 2 gives the average radius of the school, R. From Eq.(26) together with Eq.(28), the self-consistent value of the exponent α is obtained: α = 1.
(29) As a consequence, the self-consistent cohesive force reads where b denotes the effective inter-fish distance (a constant independent of N): b 2 ≡ σ 2 v /k. The average radius of the school is then given by bN 1/2 . Note that Eq. (27) is an example of a more general principle called the fluctuationdissipation theorem (Kubo, 1966).
Notice that the above model has been developed in the absence of the excluded volume interactions. Social cohesion by harmonic spring interactions between neighboring fish reminds us of the classical picture of a polymer chain based on the bead-spring model (Rouse, 1953). The exponent ν close to the value 3/5 may then be understood by taking the excluded volume effect (Doi and Edwards, 1986). Such a school without excluded volume effect may be called the "Gaussian" school in line with polymer physics.

The excluded volume effect
From Eq.(24), the probability of the school radius being between r and r + dr is given by the following (Hori, 1977): in a stationary state in three dimensions (i.e. the position vector r follows a Gaussian distribution). The Gaussian school model considered above permits fish to occupy the same region in space. Of course this is a physical impossibility since each fish possesses its own finite volume. Therefore, in the "excluded volume" school, there are a number of Gaussian school configurations which are disallowed due to the steric effect. Let p(r) be the probability that a Gaussian school configuration, as counted in Eq.(31), is also allowable under the excluded volume condition. We now calculate the probability that no overlaps occur when we place N fish within a region of volume (∼ r 3 ), which will lead to an estimation for p(r). The approach is due to Doi (1996). Letting w be the volume which is effectively excluded to one fish by the presence of another (w b 3 ), the probability that one particular fish will not overlap with another is then given by (1 − w/r 3 ). Since there are N(N − 1)/2 possible combinations of pairs, the probability that no overlap occurs in all of these combinations is given by where r 3 ≫ w and N ≫ 1. Therefore, the probability distribution of the school radius r can then be written as for the excluded volume school.
Both Ψ 0 (r) and Ψ(r) have a maximum at certain values of r. Let us estimate the average size of the school radius in each model by calculating the positions of these maxima. The maximum of Ψ 0 (r) occurs at R 0 = (2b 2 N/3) 1/2 . The maximum of Ψ(r) occurs at R, which satisfies the following equation obtained by differentiating the logarithm of Eq. (33): If N ≫ 1, the second term on the left-hand side of Eq.(34) can be neglected and hence The exponent 3/5 is the very value extracted through the data collapse.
The scaling in the relationship between geometric dimensions and biomass of pelagic fish schools is analogous to that developed in polymer physics (de Gennes, 1979). The result suggests that the dynamics are unexpectedly "simple" and depend primarily on common cohesive motion in these animate and inanimate systems. Though the above is a very rough theory of school conformation with the excluded volume effect, the overall statistical properties do not depend on the details of the model, which is a consequence of universality (Stanley, 1995).

Exploiting the analogy to a polymer chain model
As indicated in the Introduction, large internal variations in packing density occur within a school. Fréon and Misund (1999) pointed out that a source of substantial variation in internal school structure is the formation of subgroups. Such subgroups have been observed in saithe schools (Partridge, 1981), and in schools of minnow Phoxinus phoxinus (Pitcher, 1973) and herring (Pitcher and Partridge, 1979). Relatively independent movements of such clus-ters of individuals can open up empty spaces and cause large variation in school volume.
The polymer-chain analogue of the subgroup in school conformation is the "blob" (de Gennes, 1979): the polymer chain behaves as a series of blobs. Based on the blob concept, for small value of the excluded volume w, a subgroup within a school, with a number of λ B of fish, must be nearly Gaussian. We see this from Eq.(33) when we find no effect of w if wλ There is a certain value of λ B [∼ (b 3 /w) 2 ], beyond which excluded volume effects become important. A single school will appear Gaussian at scales r < r B where while at scales r > r B it will show excluded volume effects. According to the "blob" approach, an N-sized school can be described as a cluster of N/λ B subgroups. Inside the subgroup the core repulsion (by excluded volume effects) is a weak perturbation leading to a Gaussian state. The school radius is then written as R ≃ r B (λ −1 B N) ν in the native state.
Notice the plasticity of school geometry and internal structure. The compact, dense packing, as observed in artificial environments like small tanks, may be caused by the very strong stress. When the school is confronted by danger such as predators, the inter-fish distance decreases rapidly and all the vacuoles within the school collapse quickly (Fréon et al., 1992). This change may be closely related to the coil-globule transition in a polymer solution (Ptitsyn et al., 1968;de Gennes, 1975).
In real schools the nature of the short range interaction is quite complicated like van der Waals-type inter-molecular forces. Accordingly, all fish within a school interact via the two-body potential deformed from a parabolic potential of harmonic spring [grouping forces have been expressed as the gradient of a potential function in previous model studies (Mogilner et al., 2003)]. The potential change, denoted by ∆U ij , is given as the correction term to cohesive interaction potential between the i-th and the j-th fish: ∆U ij will include steric effects and also may involve weak attractions. The correction factor to the probability distribution Ψ(r) of the school radius r, in place of Eq.(32), is then given by the configuration integral where x i denotes the position vector of the i-th fish, and exp(−J∆U ij /ǫ v ) is integrated over the school configuration space. We can estimate p(r) by means of the cluster expansion (Mayer and Mayer, 1940;Uhlenbeck and Ford, 1962).
Introducing a function of separation between pairs, r ij = x j − x i , we integrate Eq.(37) in the Mayer cluster expansion over diagrams up to the second irreducible cluster, yielding with the excluded volume parameter and the other parameter defined by w △ = − ζ 12 ζ 13 ζ 23 dr 12 dr 13 .
The actual configuration is decided by maximizing the distribution function Ψ(r) = Ψ 0 (r)p(r), and this maximization condition leads to the equation determining the school radius R, Notice that, when the first irreducible cluster integral in Mayer's expansion is only taken into account, the configuration integral, Eq.(37), is evaluated by Eq.(32). In case that ∆U ij consists of a hard core repulsion and a shortrange weak attraction (i.e. at the distance of near-collision (r ∼ w 1/3 ) the attraction gradient between fish becomes steeper), Eq. (40) is estimated as w = w 0 (1 − ǫ Θ /ǫ v ), expressing the dependence of w on the velocity-fluctuation ǫ v of fish, where w 0 and ǫ Θ are constants (Doi and Edwards, 1986). During predation threats or laboratory observations, the value of excluded volume parameter w will change sign from positive to negative, provided that the velocity fluctuation ǫ v is suppressed below a certain value ǫ Θ . Decreasing ǫ v below ǫ Θ , the dimensional size of the school becomes much smaller than that of a Gaussian school, as depicted in Fig.4. According to Eq.(42) the so called "expansion factor" R/R 0 is determined not by w but by wN 1/2 , and so if N is large only a small change in ǫ v will cause a big change in dimensional size. For example, for school of 10 6 fish, a small variation in ǫ v will induce a dramatic change in the school radius. Equation (42) states that when −wN 1/2 /b 3 ≫ 1, R/R 0 is proportional to (−wN 1/2 ) −1/3 and hence the solution is as follows: Here R ′ = R/R 0 is plotted as a function of a combined variable w ′ = (9 √ 6/16) wN 1/2 /b 3 . Shown is the transition region |w|N 1/2 /b 3 ≈ 1. At ǫ v = ǫ Θ the excluded volume parameter w equals 0. The repulsive excluded volume effect then balances the attractive forces between fish, and the school behaves as a Gaussian school. The property of the swollen-dense packing transition is quite different in the two regions separated by the critical value w △ /b 6 = 0.0228. The transition between the the swollen conformation and the compact packing is gradual when w △ /b 6 > 0.0228 (dotted line), while it becomes extremely sharp and the transition is discontinuous when w △ /b 6 < 0.0228 (gray line). The solid line shows the marginal transition behavior, and the crossover takes place when w ′ = −0.242 (indicated by the arrow). Notice that the solution of Eq.(42) is multivalued [depicted by a sigmoid curve (white cutout)], showing a first-order discrete phase change (Ptitsyn et al., 1968;de Gennes, 1975), when w △ /b 6 < 0.0228. which shows the closest packing of fish in a school, i.e. the constant density independent of the number of individuals. This change may be called the swollen-dense packing transition of schools.

Configuration inside a swollen school
The approach of treating fish schools as interacting particle systems naturally leads to the idea of applying successful methods of statistical physics to the description of moving together without a leader . In statistical physics, the presence of non-trivial scaling is usually taken to mean that the dynamics are largely governed by simple geometric properties of the system and do not depend strongly on detailed properties of the system components (Wilson, 1983).
As for the school geometry, the basic units are the effective inter-fish length b and the number of fish, N. We assemble λ neighboring fish in a school into N/λ groups, e.g. in the sense of the Voronoi-Dirichlet diagram (Dirichlet, 1850;Voronoi, 1908Voronoi, , 1909. The length between centers of neighboring subgroups is defined by λνb with an exponentν. After such a scale change as the macroscopic quantities which determine the overall properties of the school satisfy g(λ −1 N, λνb) = λ χ g (N, b), where χ is an exponent which depends on the quantity under consideration. For this to hold true for arbitrary λ, the function g(N, b) must take the scaling form where a new function g (sc) is introduced. This is the Widom-Kadanoff scaling law in critical phenomena (Goldenfeld, 1992). For example, the average radius of the school, R, should be unaltered under the transformation of Eq. (44): Because of Eq.(5), R (sc) (x) = x and we must havẽ where ν = 3/5 in three dimensions.
As an example of the application of the above scaling law, let us consider the pair correlation function inside a school, where fish movement take place in a three-dimensional space. A pair correlation function ρ 2 (r) is defined as follows. We pick one fish at random in the school, and we place it at origin. Then we ask, what is the number density of other fish at a (directional) distance r from the first, and we average the result over all choices of the first fish (and of directions). The function ρ 2 (r) has an integral which is just the total number of fish per school, ρ 2 (r)dr = N . Hence, from dimensional analysis, we can write ρ 2 (r) = R −3 g(N, r/b).
Under the transformation that neighboring λ fish are grouped to form one subunit in a school, ρ 2 (r) will be reduced by 1/λ, since the pair correlation function is proportional to the number density of fish. Therefore Then, the function ρ 2 (r) obeys a simple scaling rule where ρ (sc) 2 is a dimensionless universal function, and NR −3 is the dimensional factor. Focusing on the limit r ≪ R, we can reach the form of ρ 2 (r) by a simple argument. In a sphere of radius r we have a certain number of fish,Ñ , related to r by the excluded volume exponent:Ñ 3/5 b ≃ r. The function ρ 2 (r) scales like the density of fish in the sphere, which gives ρ The possible observable property, Eq.(51), tells us that if we were to measure ρ 2 (r) for fish schools of different size N, there would be superposition of the curves obtained by plotting ρ 2 (r)R 3 /N against r/R. Thus this kind of scaling relation may be verified experimentally on large schools of pelagic fish.

Discussion
The power-law scaling generally exists in dimension-to-biomass relationship of pelagic fish schools in nature. Here I have estimated the power-law exponent ν of the geometric relation, based on the dependence of the power-law regime of the school-dimension distribution W (G) (R) on ν, i.e. Eq.(11). We have tested whether the distributions W (G) (R) are self-similar (i.e. exhibit scaling), relying on both the power law for the dimension-to-biomass relationship of Eq.(5) and the FSS relation in the power-law school-biomass distribution of Eq.(6). Plotting the scaled histogram-data from the 22 in-situ observations, we have found that the 22 curves do indeed collapse onto each other (Fig.2a), suggesting that W (G) (R) follows a universal functional form [Eq.(8)]. We have extracted the power-law exponent of dimension-to-biomass scaling relationship via a minimization of a measure to quantify the nature of scaling collapse. Next I have explained the value of the exponent ν, proposing the Gaussian school model for the fish with excluded volume w. By exploiting the analogy between fish-school and polymer conformations, we have examined the behavioral algorithm governing the swollen conformation of large-sized schools. We have seen that the exponent is modified strongly by the steric effect. What is universal in Eq.(5) is the exponent ν ≈ 0.6: it is independent of species as well as environmental conditions, and the same for all schools. The constant that multiplies N ν in Eq.(5) is non-universal [∼ (wb 2 ) 1/5 with effective inter-fish distance b, predicted from Eq.(35)], and depends on the details of interactions between fish which may vary with species and environmental conditions (Morgan, 1988).
To understand the scaling law for school geometry, Eq.(5) with ν = 3/5, it is essential to see what the value of ν represents. Obviously, for a regular object embedded into a d-dimensional Euclidean space, Eq.(5) would have the form N(R) ∼ R d expressing the fact that the volume of a d-dimensional object grows with its linear size R as R d . Contrary to an integer dimensionality, it has been shown that the packing structure within the schools is characterized by a non-integer (i.e. fractal) dimensionality of 1/ν ≈ 1.7: the number of fish in a school of radius R scales as N(R) ∼ R 1/ν . During the last decades of the twentieth century it has widely been recognized by researchers working in diverse areas of science that many of the structures commonly observed possess a rather special kind of geometrical complexity; the name "fractal" was coined by Benoit Mandelbrot for these complex shapes (Mandelbrot, 1982). Objects of biological origin are many times fractal-like (Vicsek (Ed.), 2001).
FSS plots of data lead to possibly important and practical insights in ecology.
In the context of the fisheries mission, the demonstrated geometric relation between dimensions and biomass of pelagic fish schools, when applied to mapping pelagic schooling fish, will largely improve the precision in the fish stock assessment. Because of the linear dependence of the fish population on N P , as given by Eq.(23), the fish stock abundance can be inferred from an index (i.e. cut-off size) that can be determined directly from observations [see Eq. (7)].
I am very grateful to Leah Edelstein-Keshet for helpful suggestions.