Self-Organization of Population
Structure in Biological Systems
Guy A. Hoelzer
Department of Biology
Department of Environmental and Resource Sciences
University of Nevada Reno
Reno, NV 89557
hoelzer@unr.edu
1.
Introduction
Conventional wisdom in the field of population genetics suggests that discrete boundaries
between distinctive, geographically adjacent biological populations must reflect the
influence of external factors, such as differential selection or a barrier to dispersal [Endler
1977]. Therefore, empirical observations of such boundaries are usually taken as evidence
of a previous period of geographical isolation, unless there is an obvious change in
environment that coincides with the boundary. A justification for this practice was
provided in a seminal paper [Avise 1987], which coined the term "phylogeography" and
laid the groundwork for a great deal of recent research (mostly empirical) on the spatial
distribution of genetic variation within species. However, the argument provided by
[Avise 1987] was, at best, a tenuous one that permitted research in this area to proceed
without the burden of a seriously complicating factor. They could see no reason to expect
such boundaries to emerge intrinsically and the few empirical examples available at that
time seemed to be easily explained as cases of secondary contact after a period of
complete geographical isolation [Avise 1987]. Since publication of [Avise 1987], spatial
boundaries between gene pools currently exchanging migrants on a regular basis have
been identified in many widespread species. Reliance on the argument of [Avise 1987]
has led many to infer the earlier existence of a now defunct, complete barrier to gene flow
between the gene pools. The boundaries are identified when a sufficient sampling of
individuals, locations, and genetic markers reveal relatively homogeneous regions
significantly differing in allele frequencies [Avise 1999]. Complexity theory, especially
the theory of self-organizing systems, provides a theoretical basis for emergence of
boundaries between relatively homogeneous gene pools in systems exhibiting "isolationby-distance" [Wright 1943]. Gene flow distances that are shorter than the extent of the
species' range characterize such systems. This condition affords local populations a degree
of evolutionary independence from distant parts of the species' range.
2. General qualities of self-organizing systems and their expression in
spatially-structured, biological species
The theory of self-organizing systems [Bak 1996] is still in its infancy, and the necessary
and sufficient conditions for the process of self-organization have yet to be elucidated.
However, some factors have been identified as being typical of self-organizing systems
and the logical bases for their effects have been explored [Bak 1996], particularly for
those described as complex adaptive systems [Levin 1999]. I will list some of these factors
and describe how each is expressed in the context of isolation-by-distance.
2.1. Diversity and individuality of components [Levin 1999]
Elements of complex adaptive systems have unique qualities and behave independently
(to a degree). The dynamics and structure of the system are products of the interactions
among these elements. In spatial population genetics, the distinctive elements are local
gene pools comprised of unique combinations of allele frequencies (alleles are alternative
forms of a genetic locus). Local gene pools interact by exchanging individuals through
migration and subsequent reproduction. The flow of alleles between local gene pools
limits divergence, which is advanced by localized selection, genetic drift, and mutation.
One feature of dynamic genetic systems is the constant possibility of allelic extinction.
The loss of allelic diversity would threaten the potential for self-organization, except that
mutation and recombination continually add new variants [Levin 1999].
2.2. Localized interactions among components [Levin 1999]
Isolation-by-distance geographically limits interactions among gene pools. In general, this
permits divergence among different regions of the system, which could define spatial
organization. Localization of interaction in a vast system is commonly described as flow s
within complexity theory, because it creates time lags as effects of such interactions
spread throughout the system. It is telling, or at least convenient, that the term gene flow is
used in population genetics to describe the spread of alleles across localities.
2.3. Non-linear interactions among components [Levin 1999]
Non-linear interactions in complex systems can lead to events of surprisingly large effect,
which can occur in predictable patterns, although the details of timing and causality are
not predictable for particular events [Bak 1996]. These non-linear interactions provide the
basis for the formation of boundaries among regions of the system that define its
organized structure. The boundaries themselves are non-linear outcomes, which might
require non-linear component interactions. Frequency distributions for dispersal distances
in natural systems are generally very non-linear [e.g. Wasser 1987].
2.4. An autonomous process [Levin 1999]
A process inherent to the system must cause structural organization of the system. As
[Levin 1999] pointed out, natural selection can be such a force; however, this depends on
the confines placed on the system of interest. By limiting my system to the gene pool of a
single species, many potential sources of selection (e.g. the abiotic environment and
interactions with other species) are defined as external to the system. Social interactions
among individuals, and epistatic interactions among genetic loci, could still generate
variation in selection pressures across the geographic range of a species, but this
possibility will be ignored here. Instead, I will focus on genetic drift (i.e. random changes
in allele frequencies caused by sampling error between generations) as a distinctly
autonomous process that can lead to self-organization of population substructure. In his
original paper, [Wright 1943] recognized that genetic drift acts independently at distant
locations within a system of isolation-by-distance. This means that the identities of alleles
increasing or decreasing in frequency due to sampling error are somewhat free to differ
between distant sites. Wright also concluded that isolation-by-distance reduces local
variation, while simultaneously increasing regional differences.
2.5. Dissipation [Nicolis 1989, Prigogine 1992]
Prigogine and his colleagues [Nicolis 1989, Prigogine 1992] have stressed the importance
of dissipation in self-organizing systems. He has been concerned with systems typified by
a constant input of energy, which must be dissipated before the state of maximum entropy
is exceeded, as dictated by the second law of thermodynamics. The localization of
interactions leads to viscosity of flow through the system, which makes the process of
dissipation inefficient. The flux of energy through such systems results in the formation of
structures that increase the efficiency of flow. The flow of genetic variation through
systems exhibiting isolation-by-distance is analogous to the flow of energy through the
systems described by Prigogine. Mutation provides a constant input of variation, and
population genetic flux causes old alleles to be replaced by new ones. Genetic drift results
in allelic replacement, but isolation-by-distance causes drift to be very inefficient at
purging alleles from the system once they have become widespread. I propose that the
self-organization of discrete subpopulations serves to increase the efficiency of genetic
drift as a mechanism of dissipating genetic variation. Without population subdivision,
isolation-by-distance causes species to retain more allelic variation.
Consistent with the activity of an autonomous process and continual dissipation is the
notion that self-organizing systems exhibit dynamic behavior. They are always in flux and
their general structure is actively maintained by tension between processes eroding and
regenerating structure. System dynamics, and the details of structural regeneration, are
often contingent upon unpredictable events, such as the outcome of genetic drift. The birth
and death of individuals causes genetic drift to be a continual process, and it prevents the
system from obtaining a static equilibrium state. The geographical distribution of genetic
variation is contingent on the idiosyncratic history of allele frequency changes locally, and
across the species' range. Every new generation changes the previous distribution in
unpredictable ways, so that the emergence of large-scale population structure is
necessarily a dynamic process.
3. Simulation-based evidence of the self-organization of spatial
population structure
Because isolation-by-distance has not been effectively modeled using analytical methods,
computational simulation is an attractive approach for exploration of its effects. The selforganizing property of such systems has been revealed using individual-based, spatially
explicit models in at least two instances [see also Rolf 1971]. Both explored large
geographic scales compared with average dispersal distances.
3.1. Self-organization of nuclear alleles in simulated plants
[Turner 1982] described a model in which pollination (i.e. gene flow) of individual plants
was only allowed to occur between close neighbors in a 2-dimensional lattice of 100 X
100 (population size = 10,000). To begin each simulation, two alleles, representing a
nuclear locus, were combined at random into diploid genotypes, which were then placed
onto vertices of the lattice at random. The system evolved a significant degree of genotype
and allelic spatial clumping. After 800 generations, 52% of the individuals in the system
belonged to homogeneous clumps (i.e. subpopulations without allelic variation), of 100 or
more individuals. This striking degree of population substructure emerged from the
initially random distribution of both alleles and genotypes throughout the lattice. Because
this simulation did not include the process of mutation, one allele would have eventually
reached a frequency of 100%, marking the final loss of spatial structure and the dynamic
nature of the system. Natural biological systems are constantly subject to mutation; thus
the dynamics of self-organization should be perpetual.
3.2. Self-organization of mitochondrial alleles in simulated animals
The second model to exhibit self-organization was described by [Hoelzer 1998]. It was
superficially very different from the model of [Turner 1982]. This model mimicked the
evolution of haploid mitochondrial genomes, in which alleles were not combined into 2allele genotypes, in the context of a primate social system. Individuals were organized
into social groups, which existed in a 5 X 5 lattice, and the average migration distance was
made very small by reducing the frequency with which individuals emigrated from their
natal group. Again, individuals that did migrate were constrained to enter neighboring
groups. This model included the influence of mutation, allowing for persistent system
dynamics. This model generated the same sort of clumping (i.e. spatial autocorrelation
among alleles) observed by [Turner 1982], but the transient nature of the clumps did not
doom the system to homogeneity. A small fraction of new alleles, generated by mutation,
would increase substantially in frequency and spread locally due to drift. Thus, new
clumps were continually created, which replaced those that disappeared. When average
gene flow distance was too great relative to the scale of the system, no self-organization
occurred. However, a threshold was reached as viscosity was increased, where the system
bifurcated into two subpopulations. This is biologically surprising, because dispersal
across the geographic boundary between subpopulations occurred with equal likelihood as
dispersal within the bounds of a subpopulation. The locations of boundaries between
adjacent subpopulations were arbitrary and the boundaries moved across the landscape
over time. Genetic drift was efficient within subpopulations, so little variation was found
within them at any point in time. However, the lineages occupying different
subpopulations were highly divergent. Although an analysis of the behavior of this model
was published [Hoelzer 1998, 1999], its self-organizing properties were not described.
3.3. A new simulation designed to explore self-organization of population
substructure
The simulations described above were not designed to illustrate the self-organizing
process; in fact, the observation of systemic self-organization was a surprise to the authors
of both simulations. Furthermore, each simulation had idiosyncratic features that mask the
generality of the phenomenon. Therefore, I am currently developing a new simulation
model designed specifically to study spatial self-organization, in collaboration with Chris
Ray at the University of Nevada Reno. This model includes mutation, but no social
structure. It does not require genotype or recombination analyses, because it assumes a
haploid genome. Finally, the scales of both the geographic range of the system and the
distribution of dispersal distances can be varied.
In the following simulations, a 2-dimensional lattice of 100X100 vertices was used;
thus, there was a maximum population size of 10,000, but smaller populations occurred
when some vertices were unoccupied. The lattice was rolled onto a torus, producing a
donut-shaped range without edges. Initial population size was set to 10, and each
individual had a unique mutation. Mutations color the vertex to facilitate visual
recognition of spatial genetic structure. Mutation rate was set at 105
/individual/generation. Offspring inherited the color of the parent, unless they
experienced a new mutation. Generations did not overlap, so offspring could inherit the
parental vertex. The expected number of offspring per individual was m/n, where m is a
hypothetical maximum number of individuals and n is the number of individuals currently
in the lattice. Here we set m equal to 10,000, 15,000 and 20,000 (Figures 1A, 1B, and 1C,
respectively). In each case, the actual lattice capacity was 10,000. This construction
caused the lattice to fill quickly with descendants of the 10 original founders. Isolation-bydistance was implemented by constraining the vertices occupied by offspring to either the
parental vertex or one of eight neighboring vertices. Offspring could migrate one or two
steps up, down, left, or right on the lattice. The program attempted to place each offspring
in one of these nine vertices at random, but attempts failed when the chosen vertex was
already occupied. A failed placement was followed by up to 10 new, random choices
among the same nine vertices. The method of controlling equilibrium densities described
above effectively causes the number of attempts allowed for the random placement of
offspring to vary between 10 and 20. Ultimately, failure to find an unoccupied vertex
resulted in death of the offspring. As expected, the dynamic equilibrium density on the
lattice was higher when more attempts were made to place offspring; these densities were
approximately 9,050, 9,950 and 10,000, when m was 10,000, 15,000 and 20,000,
respectively.
Figure 1. Snapshsots of simulations taken at generation 10,000. The conditions of all three
simulations were identical (see section 3.3), except for equilibrium densities, which were
about (A) 9,050, (B) 9,950, and (C) 10,000, respectively.
A snapshot of the spatial structure of genetic diversity on this landscape is shown in
Figure 1 for each of the three equilibrium densities at generation 10,000. In each case, the
widespread colors are identical to starting lineages, without any evolutionary change. The
rare, locally clustered colors represent recent mutations that have begun to spread across
the landscape. While this preliminary exploration of the model does not yet reveal
substantial subpopulations derived from mutant lineages arising during the simulation, it
nevertheless exhibits a clumped, non-random spatial distribution of colors. Contrasting
results from simulations run under different population densities reveals that competition
for space enhances the self-organizing effect. Under the high-density condition, the
parental vertex is likely to be filled by one of its offspring, but neighboring sites will
rarely be available. Thus, competition for space enhances viscosity, and the tendency for
self-organization, in a system of isolation-by-distance.
4. Implications for the field of population genetics
Contrary to the conclusions of [Avise 1987], and current standards of practice in the field
of phylogeography, observations of such boundaries in natural populations do not
necessarily indicate secondary contact or the effects of selection in different
environments. The model described here suggests that such boundaries can emerge as a
result of the internal dynamics of a system exhibiting isolation-by-distance. Therefore,
this model provides a new null hypothesis, which predicts the occurrence of boundaries
among subpopulations, and the maintenance of highly divergent alleles without
intermediate forms, in natural populations that are sufficiently viscous relative to their
geographic ranges. This is a null hypothesis, because it does not attribute the pattern to the
influence of any factor external to the system under study. Unfortunately, this makes
study of such external factors more difficult.
4.1. If a boundary is not evidence of secondary contact or local adaptation, then what
would constitute evidence of these phenomena?
The influence of external factors on spatially structured populations was a subject of
investigation before the advent of phylogeography. The model presented here brings into
question the validity of some uses of phylogeographic methods, but it does not impinge on
more traditional approaches. For example, physical (e.g. fossilized) evidence of historical
ranges can suggest secondary contact, and local adaptation on either side of a boundary
can be studied experimentally (e.g. through reciprocal translocations). Indeed, observation
of a phylogeographic boundary can be the basis for hypotheses about the roles of external
factors, which can then be tested in these ways. It is also possible, given enough data, that
this null hypothesis could be rejected based on predicted patterns of system dynamics. For
example, complex systems are generally characterized by fractal patterns, 1/f noise, and
power law relationships [Bak 1996]. While it has yet to be determined where the power
laws lie in the population genetics of isolation-by-distance (perhaps the ranked frequency
distribution of alleles at any point in time?), I expect such relationships to be predicted by
this null hypothesis. The influence of external factors might make some of these
predictions false.
4.2. Future directions
Exploration of the range of conditions under which spatially structured population
subdivision is expected to self-organize will be needed to appreciate the potential role of
this process in natural systems. It is possible that the ratio of gene flow distance to range
required for self-organization is rarely realized in natural systems. It is also possible that
external forces frequently interfere with the self-organizing process. Following further
development of the theory, empirical research will be needed to explore these
possibilities. However, before this research program is initiated the possibility of
population self-organization must first be appreciated. I expect that this will require a
period of intellectual digestion, including some discomforting indigestion, because
biologists are traditionally trained to look for the influences of external factors when
systemic structures are observed. Appreciation of this model will necessitate a new,
general way of thinking about problems for many biologists.
I also expect that natural systems will not fit cleanly into either the model of selforganizing systems or the traditional view that all structure is explained by the effects of
external forces. It is likely that both internal and external sources of structure interact in
nature. For example, this null model predicts that the locations of boundaries will be
arbitrary, assuming environmental homogeneity. This means that the model also predicts
that boundaries existing for unlinked genetic markers would not necessarily coincide;
however, features of the landscape might marginally reduce local dispersal and attract
otherwise arbitrarily located boundaries. Geographic heterogeneity of this sort might
cause alignment of boundaries for unlinked markers, resulting in a pattern of population
subdivision reflecting most of the genome. This could set the stage for parapatric
speciation [Endler 1977] in a way that has previously been unappreciated.
References
Avise, J.C., Arnold, J., Ball, R.M., Bermingham, E., Lamb, T., Neigel, J.E., Reeb, C.A., &
Saunders, S.C., 1987, Intraspecific phylogeography: the mitochondrial bridge between
population genetics and systematics, Annual Review of Ecology and Systematics, 18, 489-522.
Avise, J.C., 1999, Phylogeography: The History & Formation of Species, Harvard University Press
(Cambridge).
Bak, P., 1996, How Nature Works: The Science of Self-Organized Criticality, Springer Verlag (New
York).
Endler, J. A., 1977, Geographic Variation, Speciation, and Clines, Monographs in Population
Biology, no. 10, Princeton University Press (Princeton).
Hoelzer, G. A., Wallman, J., & Melnick, D. J., 1998, The effects of social structure, geographical
structure and population size on the evolution of mitochondrial DNA. II. Molecular clocks and
the lineage sorting period, Journal of Molecular Evolution, 47, 21-31.
Hoelzer, G. A., Wallman, J., & Melnick, D. J., 1999, Erratum: The effects of social structure,
geographical structure and population size on the evolution of mitochondrial DNA. II. Molecular
clocks and the lineage sorting period, Journal of Molecular Evolution, 48, 628-629.
Levin, S., 1999, Fragile Dominion, Perseus Books (Reading).
Nicolis, G., & Prigogine, I., 1992, Exploring Complexity, W. H. Freeman (New York).
Prigogine, I., 1992, Dissipative structure in quantum theory, Quantum-Theory Physics ReportsReview Section Of Physics Letters, 219, 93-108.
Rolf, F. J., & Schnell, G. D., 1971, , American Naturalist, 105, 295-324.
Turner, M. E., Stephens, J. C., & Anderson, W. W., 1982, Homozygosity and patch structure in
plant populations as a result of nearest-neighbor pollination, Proceedings of the National
Academy of Sciences USA, 79, 203-207.
Wasser, P. W., 1987, A model predicting dispersal distance distributions, in Mammalian Dispersal
Patterns: The Effects of Social Structure on Population Genetics, edited by B. D. Chepko-Sade
and Z. Tang Halpin, University of Chicago Press (Chicago).
Wright, S., 1943, Isolation by distance, Genetics, 28, 114-138.