Download Since the detection of genes as units of heredity, the nature

Non-linear self-organizing epigenetic processes as
a third source of developmental differences:
implications for genetic analysis
Universiteit van Amsterdam
Faculteit der Psychologie
Studieonderdeel: Scriptie
Datum: 19 Mei 2006
Auteur: Kees-Jan Kan
Studentnummer: 9881379
E-mail: [email protected]
Begeleider: drs. A. Ploeger, dr. M. E. J. Raijmakers
Abstract
Behavior geneticists traditionally explain phenotypic variation as a result of genotypic
differences and environmental influences acting upon it. The latter are usually subdivided
into within-family and between family influences. Unfortunately, according to some
empirical data a substantial portion of phenotypic variance can’t be explained by this
model. It is argued that non-linear epigenetic processes, capable of self-organization,
constitute a third factor of phenotypic differences in development. The possible role of
chaos theory and its implications for the genetic analysis of phenotypic variation is
illustrated by applying the discrete logistic function as a model for development. Finally,
a twin simulation study employing a more biologically plausible model, e.g. for neuronal
network development, is proposed.
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Contents
1 Introduction
4
2 Empirical evidence of a third component causing random variability
7
3 Autonomous epigenetic processes as a third source of developmental differences 11
4 The discrete logistic equation as a simple model of development
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5 Conclusions
22
6 References
24
3
1. Introduction
Since the detection of genes as units of heredity, the seemingly never-ending naturenurture controversy took its recent form of the genes-versus-environment debate. The
essential question of the debate is whether biological and behavioral individual
differences are primarily resulting from genetic differences or from environmental
differences. In investigating and understanding these individual differences and their
sources of variation a few concepts are essential. Two of the most important concepts are
the concepts of phenotype ( P ), which refers to apparent, observable, measurable
characteristics of the individual, and the concept of genotype ( G ), which refers to the
unobservable, latent characteristics of the same individual (Grigorenko, 1999).
Traditionally, (behavior) geneticists make this distinction, while it is assumed that
the difference between phenotype and the underlying genotype is influenced by the
environment ( E ). Usually the environmental influences are subdivided into shared ( E1 )
and non-shared ( E2 ) environmental influences (e.g. Bouchard & McGue, 2003;
Molenaar, Boomsma, & Dolan, 1993). Shared environmental influences result in family
members becoming more alike for a given trait. In contrast, non-shared environmental
influences make them different.
This model can be expressed in quantitative terms, including the interactive term
( G × E ), referring to possible combinations of genetic and environmental effects, as
follows,
P = G + E1 + E2 + (G × E )
With variance,
VP = VG + VE1 + VE 2 + 2Cov(G )( E ) + VG×E
where 2Cov(G )( E) denotes the covariance between genetic and environmental effects.
One example of such gene-environment correlations is the fact that one person’s own
characteristics may have evocative influences on the ways other people respond. An
example for the other type of interplay between genes and environment, interaction, is
that only some people are genetically susceptible to a particular allergen. As long as the
environment doesn’t contain the allergen, the individual won’t be affected; non-
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susceptible individuals will be free from the allergic effect even in an allergenic
environment (Rutter et al.,1997).
Besides genotype and phenotype, another central concept in behavior genetic
2
research is important: the population statistic Heritability ( h ). Heritability is defined as
the proportion of phenotypic variance in a particular trait attributable to genetic variation
at a given time (Gray, 1999). As a formula, it is written.
h2 =
VG
VP
To estimate the heritability of a trait, correlations between certain types of close
relatives can be compared. One method of determining each of the contributing
components of the phenotypic variance, and eventually the estimate of h 2 , is the
selective breeding of organisms for a given trait. Subsequently, the observed response to
genetic selection can be explored. However, for obvious ethical and moral reasons,
scientists cannot conduct such genetic experiments on humans. The same argument holds
for the recently available method of cloning organisms. But, fortunately, researchers
investigating human heritability can utilize “experiments done by nature” and assess the
similarity between relatives.
Twin studies are an especially valuable tool in behavior genetics. Monozygotic
(MZ) twins share 100% of their genes, whereas dizygotic (DZ) twins, just as regular full
siblings, share, on average, only 50% of their genetic makeup. Moreover, they can share
the environment in which they are raised or can be adopted at an early age in different
environments. By comparing sets of correlations ( r ) between MZ and DZ twin pairs it is
able to estimate the relative contributions of genetic and environmental variance
components (Plomin, Defries et al., 2000),
rMZ = h 2 + c 2
rDZ =
h2
+ c2
2
where c 2 is the proportion of shared environmental variance. Solving the equations
yields,
h 2 = 2(rMZ − rDZ )
5
c 2 = rMZ − h 2
Knowing that h 2 , c 2 and e 2 (the relative contribution of non-shared environmental
variance) sum up to 1, while they represent proportions, the formula for e 2 yields,
e 2 = 1 − rMZ
In other words, any variance that is not shared between MZ twins must be due to nonshared environmental sources of variance. It is important to note that, as a result, the error
variance in this model, partially due to imprecise measurement of the phenotype, is
automatically subsumed under the non-shared environmental component (Grigorenko,
1999).
Despite the seeming plausibility of this standard quantitative genetic model there
remains an unresolved random variability in the process of development. Several authors,
including Molenaar, Boomsma, & Dolan (1993), Finch & Kirkwood (2000), Gärtner
(1990), Smith (1993), Lajus et al. (2004), have suggested the presence of a third source
creating phenotypic variation, in addition to genetic and environmental factors. In this
paper it is argued that this source requires separate and explicit recognition in order to
resolve the nature of interindividual differences.
In the next section an extensive list of empirical evidence of the existence of this
third source of variance is examined, followed by theoretical considerations. Also,
simulation studies of twin pairs by Eaves et al. (1999) and Molenaar & Raijmakers
(1999) are presented, and their implications for behavior genetic modeling are discussed.
Finally, some suggestions are made for a twin simulation study, applying a realistic,
biological plausible model (see Van Ooijen, van Pelt & Corner, 1995) for neuronal
network development.
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2. Empirical evidence of a third component causing random variability
As pointed out in the introduction there is empirical evidence, from both animal and
human studies, suggesting that some phenotypic differences can’t be explained by
genetic, environmental factors, or their interactions alone. One of the pioneers in
addressing this question is the German researcher Gärtner.
Gärtner (1990) reviews his own laboratory’s decades-long efforts to reduce the
variability of biological traits in laboratory animals by standardization of environmental
conditions and genotypes, initially in order to increase the validity of experiments. Using
highly inbred rats under strict environmental control (standardization of components as
food, temperature, group size, etc.) did not significantly decrease the randomly
distributed variation in a number of quantitative traits, such as kidney weights, when
compared with a group of rats living in a natural, wild setting. Apparently neither the
postnatal environment nor genes did constitute the major source of phenotypic variation.
Subsequent research was performed to measure the environmental variability
directly. For this purpose eight-cell stage mice embryos were divided, thereby creating
monozygotic twin pairs. Each twin pair was transplanted into the uterus of the same
foster mother, which raised both twins. Several physical characteristics, including body
weight for example, were measured after birth and compared with those of natural born
mice. The coefficients of variation appeared to be similar in both groups and it was
estimated that environmental influences caused only 3-30% of the variability.
In a similar experiment with Friesian cattle, an additional comparison was made
with a group of divided embryos, transferred into and raised by different uterine foster
mothers. As well as in the previous described experiments, a large amount (70-97%) of
the random phenotypic variability remained unexplained. From these studies, the
conclusion was drawn that the remaining variance had to be attributed to a third
component, in addition to genetic and environmental influences. According to Gärtner
(1990) this component must be inborn and may stem from (molecular) cytoplasmic
differences in the zygote.
Since the recently technique of cloning animals is available (another technique of
creating isogenic individuals), results in concordance with those of Gärtner (1990) are
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demonstrated. For example, Archer et al. (2003) compared cloned pigs on several
physical and physiological traits with naturally bred controls. Controls were matched for
age, breed and sex and were held under identical conditions. Body weight, number of
teats, hair growth pattern, skin type, and blood parameters, such as alkaline, glucose and
calcium, were measured. Analysis of the degree of phenotypic variation between clones
and controls indicated the existence of two classes of traits. In one class of traits cloned
pigs showed, as expected, less variation in relation to controls. However, in the second
class the cloned animals displayed equally high or even increased variability. Although
environmental conditions can’t be controlled fully, it is highly unlikely that
environmental effects could account for physical traits as, for instance, hair growth
pattern or skin type. The experimenters suggested that these differences in traits could be
caused by small deviations introduced during cloning, microenvironmental influences or
minimal initial differences in uterine conditions.
Besides phenotypic variation in genetically identical animals, another kind of
variation reflects the incomplete ability of organisms to develop the same phenotype
under the same environmental conditions. Bilaterally organisms can show small, random
deviations in morphological structures between the right and left side. This intraindividual type of variation is known as fluctuating asymmetry and is found in various
traits in various animals. For an example, Stige et al. (2005) repeatedly measured feather
length and color pattern of wings and tail of the pied flycatcher, a type of bird. The values
of feather asymmetry persisted from nestling stage to adulthood, even across moults.
These findings suggest high heritability, while the asymmetry is determined early in life
and is largely unchangeable. However, genetic analyses revealed heritability was almost
non-existent and shared environmental factors had little influence. Moreover, differences
in within-nest conditions couldn’t explain the random variation in fluctuating asymmetry.
The researchers concluded asymmetry is possibly determined by stochastic events during
early stages of development, permanently affecting the feathers. In addition, according to
Molenaar et al. (1993) (referring to a study of Mather & Jinks in which variation in
numbers of chaetae between right and left sides of inbred fruit flies was found), it is not
likely that the environment systematically influences the left and right side in a different
way.
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Fluctuating asymmetry is not limited to animals. In humans researchers observed
small morphological asymmetries in various bilateral traits as well; in tooth size, skin
patterns on the palms of the hands, ankle and foot breadth, elbow breadth, hand and wrist
breadth, ear length and breadth, finger lengths, and facial features (see Kowner, 2001).
Kowner too, puts forward that these bodily and facial variations emerge at a very young
age, remain stable throughout the lifespan, and the direction is not under genetic control.
A similar point of view is given in a study by Fink et al. (2004), in which it is
furthermore argued that facial asymmetries seem to be the results of instabilities during
the first stages in embryonic development.
A related phenomenon, and another indication of developmental instability, is the
development of so-called minor physical anomalies, small deviations in structural
features of no functional significance (Kowner, 2001). The study of Townsend et al.
(2005) illustrates nicely the role of developmental instability in creating phenotypic
variability. These researchers examined the dental records and radiographs of 278
monozygotic twin pairs and found that 24 pairs had missing or extra teeth. However, 21
of these 24 pairs showed between-pair differences in patterns of expression. For example,
one pair missed a premolar, one twin on the left side, the other on the right, showing a
mirrored effect. In their view, the observed differences in dental features of monozygotic
twins might stem from molecular interactions, leading to the initiation and ultimately the
differentiation of developing teeth.
So far, only studies investigating biological traits have been discussed, however
there is no valid argument against the hypothesis that processes as described above could
influence the organization and structure of the human brain. In fact, several studies (e.g.
Thoma et al., 2002; Wright et al., 2002; Eckert et al., 2002; Steinmetz, 1996) have
demonstrated the existence of atypical, non-genetically determined, asymmetries in
human brain anatomy. Eckert et al. (2002) examined the heritability for asymmetry in the
planum temporale in 27 monozygotic and 13 dizygotic male twins. Magnetic Resonance
Imaging (MRI) measurements revealed significant dissimilarities in gyral and sulcal
features between monozygotic twins. The authors hypothesized these intra-uterine effects
could lead to divergent morphological development. When twins with birth weight
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differences were excluded from genetic analysis the magnitude of asymmetry decreased,
giving stronger evidence for such prenatal influences.
Modern scientists believe that differences in brain structure can lead to
differences in brain function and, since behavior is a reflection of brain function, can
ultimately result in differences in manifest behavior (Benno, 1990, p.114). As mentioned
earlier, there remains an unresolved variability in the process of development. This
applies not only for (neuro-) biological data, but for behavioral data as well. For
personality, intelligence and psychopathology, a substantial part of non-genetic variance
has to be attributed to non-shared environmental influences, at least according to the
classical behavior genetic model (Molenaar et al. 1993; Smith, 1993).
Smith (1993) concentrates on a number of twin and familial studies investigating
genetic and environmental contributions to intelligence. One of these studies is a study by
Vroon, de Leeuw and Meester, who investigated the relationship in intelligence between
sons and fathers in Holland, examined for military service at age 18. Scores of 2847
father-son pairs on the Raven Progressive Matrices test were available. Data analysis and
path analysis led to the conclusion that only 3% of the variation in son's IQ was explained
by IQ of the father and educational variables. Smith (1993) argues that some part of the
variance in personality traits ascribed to the non-shared component is certainly due to
errors in measurement, but it’s implausible to suppose measurement error accounts for an
amount of variance as large as the variance caused by genetic and/or environmental
factors. In the end, he concludes the intrinsic dynamics of the brain (indirect influences of
spontaneous synaptic changes in activity) partially explain developmental variability.
In sum, experiments with inbred and isogenic animals, studies of developmental
instability in both humans and animals on bodily and brain morphology, as well as twin
and familial studies present convincing evidence of the existence of a third component, in
addition to genetic and environmental influences, in causing phenotypic variability in
development. This component may consist of (stochastic) molecular or cellular processes
creating seemingly random intra- and inter-individual phenotypic differences at both
somatic and behavioral levels, and is subsumed under the non-shared environmental
component in the standard behavior genetic model. In the next section some theoretical
considerations by Molenaar et al. (1993) are introduced.
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3. Autonomous epigenetic processes as a third source of developmental differences
As described above, there is convincing empirical evidence indicating the existence of an
independent source of phenotypic variation, in addition to genetic and environmental
factors. Molenaar et al. (1993) argue that this third source may consist of autonomous
non-linear epigenetic processes, having self-organizing properties. These endogenous
processes are deterministic, yet give rise to chaotic output under constant genetic and
environmental conditions. In order to understand what is implied by such a process, a few
definitions and concepts have to be given or explained.
First of all, it is useful to explain what is meant by chaos. In a meta-physical
sense, chaos is the opposite of law and order. In a more scientific sense, chaos refers to
some specific behavior found in certain nonlinear dynamical systems. A dynamical
system is a system that changes over time. Dynamical Systems Theory, or simply
Dynamics, refers to the study of these systems in the process of change1. Dynamics deals
with mathematical objects that unambiguously describes (i.e. specified by a set of
differential or difference equations) how the state of the system evolves over time (Beer,
2000, van Gelder & Port, 1995)). A dynamical system can change over time in either a
1
In conceptualizing how systems change over time, dynamical system theory employs geometrical
representations. A system’s state corresponds with a point in an n -dimensional space ( n denotes the total
number of variables), the state space. The path through which the system traverses over time is known as
its trajectory or orbit and can be represented by a continuous curve or a sequence of points, depending on
how time is defined; respectively in real time or in discrete time. Starting from some initial state, the state
of many dynamical systems ultimately ends up into a small subset of the state space known as limit set or
attractor (Bechtel & Abrahamsen, 2002). Equilibrium points and limit cycles are two examples of simple
types of stable attractors producing constant, respectively endless rhythmic behavior. For stable attractors,
all nearby trajectories converge to the attractor, so that small perturbations away from it will return to the
identical attractor. However, some attractors are unstable, so that any perturbation away from it will drive
the system away from the attractor (Beer, 2000). One example of such an unstable attractor is a chaotic
trajectory, which means the bounded trajectory never repeats itself, and thus appears random. Yet, the
trajectory is actually following a deterministic path, i.e. the next state of the system depends on the current
state, fully determined by an algorithm (Beer, 2000; Bechtel & Abrahamsen, 2002). For that reason the
functioning of such systems is generally known as deterministic chaos.
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linear or a non-linear way. Linearity essentially means that effects are proportional to
their causes. In contrast, the essence of non-linearity is that small causes, initial changes
or differences, may result in large consequences. Furthermore, the presence of nonlinearity implies the possibility of very rapid changes, especially when the system in
question is a feedback system. Chaos refers to these phenomena and can be defined as the
unpredictability inherent in a system in which apparently random changes occur as a
result of the system’s extreme “sensitive dependence on initial conditions” (Barton, 1994;
Warren et al., 1998)). In popular science this phenomenon is named the butterfly effect:
In theory a butterfly flapping his wings on one side of the earth can cause a heavy storm
on the other side. In general, the weather is a good example of a chaotic system.
It is important to note that the unpredictability of the system arises from the lack
of absolute precision in the initial conditions (i.e. the values of the measurements at a
given time) and not from any randomness in the equations, while they are completely
deterministic, hence the random behavior is only apparent. In practice, since no
measuring device can record its measurement with infinite precision, the measurement
always includes a degree of uncertainty in the value. Even in theory, exact starting values
of real processes would require an infinite amount of energy, which is physically
impossible (Molenaar & Raijmakers, 1999). Hence, two nearly-indistinguishable sets of
starting values for the same system could result in two extremely different predictions.
Therefore, in the example of the weather as a chaotic system, predictions for the weather
are only possible in the short-term. In general, in studying real chaotic systems long-term
predictions are both theoretically and practically impossible to make, neither in advance
nor in retrospect, or would be as inaccurate as any random chance prediction.
A second notion concerns the non-linearity (the origin of sensitivity dependence
on initial conditions) of some systems, the corresponding mathematical models and their
solutions. Linear systems are most effectively modeled by a set of linear equations, which
are additive. That is, solutions can be combined to obtain another solution. Non-linear
systems, modeled by non-linear equations are not additive and therefore usually difficult
to solve. Often, a single solution can’t be obtained and a pattern of solutions constitutes
the answer. In finding such a pattern, a process called iteration is generally used: a data
set is put in the set of equations and the ultimate output is fed back into the equation set
12
as new input and the whole process is repeated. Sometimes a bifurcation occurs in the
pattern of solutions: a sudden jump from one set of solutions to the next. For example,
such a transition is found in finger tapping. If one alternately taps the index fingers and
increases the frequency gradually, one suddenly jumps into in-phase tapping once a
certain limit frequency is exceeded. A small, but critical, smooth change made to the
parameter values of a system causes a sudden qualitative change in the system's long-run
stable dynamical behavior. In a bifurcation diagram, the geometric representation of the
pattern of solutions, a bifurcation can be recognized as a splitting of the solution.
Ultimately, the solutions can jump to very complex chaotic behavior. (Warren et al.,
1998; Barton, 1994).
Surprisingly, chaotic systems can attain certain levels of order, structure, and
stability. This process is known as self-organization. Camazine (2001, p.8) provides the
following definition: “Self-organization is a process in which pattern at the global level
of a system emerges solely from numerous interactions among the lower-level
components of the system. Moreover, the rules specifying interactions among the
system’s components are executed using only local information, without reference to the
global pattern.” Here, pattern (or structure) refers to a particular organized
spatiotemporal arrangement of objects; something nonrandom in form, something that
recurs in time, or both. It is an emergent property of the system, formatted through
internal interactions of the system. That is, the order is not imposed by external shaping
forces. Hence, a self-organizing process is autonomous and self-regulating. To be able to
generate these highly structured patterns from homogeneous initial states the system must
satisfy a few conditions (Bonabeau et al., 1997; see also Meinhardt, 1982). First, strong
positive feedback (autocatalysis) must occur in order to promote changes in the system.
Positive feedback reinforces change in the same direction as the initial change, so that
perturbations are amplified. Second, the short-range positive feedback must trigger and
be coupled with negative feedback (lateral inhibition). Negative feedback will serve as a
stabilizing mechanism by promoting change in the opposite direction to a perturbation.
Without negative feedback the structure would become amorphous. Third, the system
must involve large numbers of components, so numerous actions and interactions will be
possible. The rules specifying the interactions are based upon local information. Fourth,
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randomness (external or internal fluctuations, perturbations, errors, noise) is crucial,
generating local heterogeneities in structure which become amplified by positive
feedback. Furthermore, self-organizing systems are open systems and require an influx of
energy, matter or information from the environment.
Self-organization and pattern formation are found in both living and non-living
systems (Meinhardt, 1982; Camazine et al., 2001). A little stone can serve as a small
wind shelter; thereby producing a sand deposit; this deposit increases the wind shelter
and more sand is deposited. The process continues and accelerates until nature is out of
resources or some physical law is playing the limiting factor. The ultimate result is a
large sand dune. Other examples of patterns and pattern formation in the physical world
are, among others, spiral waves produced by the Belusov-Zhabotinsky chemical reaction,
star and galaxy formation, snow flakes, cloud streets, ocean waves and hurricanes. In
plants one can find the branching structure of trees, the venation of leaves, the pattern of
leaves in a cross-section of red cabbage, and the head of a morel mushroom. Well known
examples of patterns believed to involve self-organization are found on skins, coats and
shells of various animals: zebras, giraffes, tigers, leopards, snakes, seashells, etc. Patterns
might be the result of animal behavior; a colony of termites is able to build complex nests
and bees develop honey combs. Also animal behavior can exhibit structure in time or
space; the movements of a flock of birds, fish swimming in schools and synchronized
flashing of fireflies are typical examples.
In animals, including humans, examples of self-organization have been described
in the process of morphogenesis (Meinhardt, 1982; Camazine, 2001) the development of
tissues, organs and overall body anatomy. During, embryogenesis, the embryonic
development of an organism, organized spatial distributions of cells arise, which give rise
to the typical structures. Skin and coat patterns as mentioned above, and also the
formation of vertebrate limbs and segments of insects are, at least in part, the result of
self-organizing processes. Organs, such as lungs, display a branching structure, as well as
the cardiac muscle network and the blood circulatory system. Last but not least, the brain
is a highly spatial organized neural network.
Self-organization may provide an explanation for the puzzling fact that the total
amount of information stored in the genome is far too small to describe the structure of
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the adult brain (Molenaar et al, 1993; Benno, 1990), not to mention the entire individual
(Camazine, 2001). Genes only need to carry a set of rules to generate that information.
Stated somewhat differently, self-organizing processes are considered to be a factor in the
ways genotypes develop into phenotypes. Here, the terms epigenetics and epigenesis
come into play. Epigenetics, a term invented by C.H. Waddington, originally referred to
the study of epigenesis, “the way genes and their products bring the phenotype into
being” (Jablonka & Lamb, 2002). Waddington visualized the developmental system as a
landscape (the epigenetic landscape2) in which multiple bifurcating and deepening
valleys and hills run down from a plateau, which topology is determined by the genetic
makeup (Saunders, 1993; Jablonka & Lamb, 2002). An organism’s dynamic development
from genotype to phenotype can be seen as a ball making its way downhill. The course of
development is able to follow many, but limited, pathways, or chreods. On the one hand,
branches may rejoin further on, so that different paths will nevertheless lead to the same
phenotype (canalization). On the other hand, certain paths will branch off from one
another to be separated by hills so that different phenotypes can be reached from the
same starting point (plasticity). Environmental influences, as well as external and internal
perturbations tend to push development from its normal pathway (or may bring about a
change in the landscape). Depending on the magnitude of the disturbance and the
steepness and deepness of the valley, development may return to the original pathway or
will be diverted to an alternative pathway. One can imagine that especially at branching
points and early in development, even small perturbations may have large consequences
in the end.
Through time, epigenetics changed in the meaning given, and at present it is used
in a different sense than Waddington applied it (Jablonka & Lamb, 2002; Haig, 2004).
During the 1990’s the scope was narrowed further and further while the understanding of
the molecular mechanisms that control gene activity during embryonic development and
cell differentiation increased. Epigenetics became defined as “the study of mitotically
and/or meiotically heritable changes in gene function that cannot be explained by changes
2
Clearly, the epigenetic landscape represents a class of dynamical systems. In Dynamical Systems Theory
terms, the landscape itself is the state space, phenotypes are attractors, and chreods are trajectories.
Reversely, Waddington called the state of the system epigenotype.
15
in DNA sequence” (van Speybroeck, 2002; Haig, 2004). Despite differences in meaning
and explanation, both the Waddingtonian and the molecular biological sense focus on
alternative developmental pathways and on the influence of environmental conditions and
their consequences for the organism.
As an example in line with this modern sense of epigenetics, consider the study of
Fraga et al. (2005), in which the authors indicate the existence of epigenetic differences
as one possible explanation for phenotypic discordances observed in monozygotic twins,
such as susceptibilities to disease and differences in anthropomorphic features. The
global and locus-specific differences in DNA methylation and histone acetylation of a
large number of MZ twins were examined. The results demonstrated that MZ twins are
epigenetically indistinguishable during early lifetime, whereas older twins display
substantial differences in the patterns of DNA methylation and histone acetylation. A
possible and partly explanation of this divergence is that small defects in transmitting
epigenetic information through successive cell divisions accumulate during aging. This
process, known as epigenetic drift, may have an important impact on gene expression,
and thus on phenotype. According to the researchers these findings also support the
differences found in the discordant frequency/onset of diseases between MZ twin pairs.
At this point, after a highly modest introduction in chaos theory, self-organization
and epigenetics, one can imagine how an organism (a non-linear dynamical system)
possibly develops from genotype to phenotype. After fertilization, the non-linear
mechanisms, determined by the genetic makeup, give rise to autonomous self-organizing
growth processes leading to structure and patterns. However, the environment, the selforganizing process itself and their interactions with each other and with the genes tend to
perturb development away from the original pathway. At critical points (bifurcation
points), these perturbations may direct development following a different pathway,
resulting in an alternative structure or pattern at both somatic and behavioral levels.
According to Molenaar et al. (1993) it are these autonomous self-organizing
processes during epigenesis that constitute the independent third source of developmental
differences. Even under constant genetic and environmental influences these processes
are able to create variability. The authors propose that if such a process would be
computer simulated twice, where all initial conditions, genetical influences and
16
environmental influences would be identical, still different outputs would be obtained. In
general, the application of the relative recent available technique of computer simulations
is an interesting and welcome supplement in developmental psychology and particularly
in the discussion of the third source of phenotypic variability as the next section will
illustrate.
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4. The discrete logistic equation as a simple model of development
As pointed out in the previous section, intrinsic, non-linear epigenetic processes can
result in chaotic time courses and might be a source of both intra- and inter-individual
phenotypic variability. Eaves et al. (1999) present twin pair simulation studies using a
simple non-linear developmental model capable of producing such chaotic or nearchaotic behavior. MZ and DZ twin time series were simulated under various assumptions
concerning starting value and parameters. Following Eaves et al. (1999), Molenaar &
Raijmakers (1999) carried out a simulation, using the same model, but applying an
alternative design.
Eaves et al. (1999) consider the behavior of the logistic equation
xn +1 = kxn (1 − xn )
on the unit interval [0,1] as a simple non-linear model for development. The current value
of x ( xn +1 ) depends on its previous value ( xn ), and, ultimately (after n iterations), on the
starting value of x ( x0 ). The value of parameter k and the form3 of the logistic equation
were considered to model the epigenetic process of trait x , the observed phenotype. In
each of the series of simulations the phenotypic correlation between 500 twin pairs was
calculated for a total of n = 50 iterations. Starting values were normally distributed with
mean µ and standardized total variance (between-pairs ( σ w2 ) + within-pair variance ( σ b2 )
= 1).
3
The behavior of the logistic equation can display dramatic differences, depending on the value of k and,
in some cases, on the value of x0 . For values of k ≤ 3 x converges to a fixed value, irrespective of the value
of x0 . For values of k ≤ 1 the equilibrium state is always 0. For values of 3 < k < kcrit ( kcrit
≈ 3.5699 )
x will oscillate between 2 or more bifurcation points, and x0 has some influence on the order of linear and
chaotic periods. Above values of kcrit , the critical value of k , the equation shows true chaotic behavior and
small initial differences in starting values can lead to extreme differences in the sequential course of the
function. For values of k > 4 , the iterated value of x may lie outside the unit interval, with the consequence
that the value of x will approach infinity after further iteration. From an epigenetic/behavior genetic point
of view, only values of 3 < k ≤ 4 were used in the simulations.
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When the values of xn +1 (with parameter values of k = 3.9 and x0 = 0.5 for
example) are plotted against the series of iterations, the points are aperiodic and hence
appear random, yet they are the result of a deterministic mathematical function. The
hidden order reveals itself when xn +1 is plotted against xn (autocorrelation) and all of the
apparently random output from the logistic equation get arranged and fall along a strange
attractor. In this case the strange attractor is shaped as a parabola (Warren et al., 1998).
From the simulations of Eaves et al. (1999) it appeared that for the applied values
of k = 3.4 , as well as for values of k = 3.84 , correlations remain relatively high during
the process of iteration for small within-pair variances, whereas for higher within-pair
variances the correlations decrease rapidly. For a value of k = 3.4 the convergence
occurs in a relative smooth fashion, compared with the oscillating convergence, to
approximately the same limiting values, for a value of k = 3.57 . A greater value of
k ( k = 3.84 ), causes a rapid decrease in phenotypic correlation coefficients during
iteration, regardless of the size of the within-pair variance.
Eaves et al. (1999) draw some specific conclusions from their simulation
experiments. Firstly, they conclude that, as a result of non-linear epigenetic processes
small variations in initial conditions can have considerable phenotypic consequences
which may look like occasion-specific environmental effects. Thus, under a linear genetic
model researchers may be vainly looking for (short term) exogenous variables explaining
their data, whilst in reality no environmental influences are being effective. Secondly, for
genetic effects a similar argument may apply. That is, under the classical linear model
one might be inclined to presume for example that specific loci are responsible for a
genetic effect during the process of development, whilst in fact non-linear epigenetic
processes are accountable. Finally, they conclude that when in twin and family research
correlations between dizygotic twins are found to be approaching zero, whilst
correlations between monozygotic twins remain high, these findings form an indication
of the presence of non-linear developmental processes. According to Eaves et al. (1999),
taking the actual developmental data in consideration, these processes do occur (As
demonstrated by EEG spectra for example), yet these occurrences are relatively rare and
may not be universally present. In other words, the suggestion is made that chaotic
processes play just a rather limited part in the development of an individual. By
19
discussing the design and results from their own twin simulation studies, Molenaar &
Raijmakers (1999) argue that this suggestion needs further qualification.
Whereas Eaves et al. (1999) kept the values of the parameter k identical for each
individual, Molenaar & Raijmakers (1999) let this parameter vary under complete genetic
control (heritability = 1) with mean k = 3.8 and a between-subjects variance of 0.01. In
addition, variation in the starting value was induced under near complete genetic control
(heritability = 0.999999) with mean x0 = 0.5 and a between-subjects variance of 0.01.
Within-pair correlations were calculated for 100 MZ and 100 DZ simulated twin pairs
over a total of 200 iterations. From this alternatively designed simulation it was found
that after a finite time interval, the within-pair correlations of both MZ and DZ pairs will
decrease significantly to about zero, although at different rates: DZ twin-pair correlation
decrease almost immediately, whilst MZ within-pair correlation is high at an early stage
and decreases just at a later stage and more slowly.
Applying standard behavior genetic analyses yields different estimated
heritabilities and different fits to the data for separate time intervals. During an early time
interval the model yields high heritability estimates and acceptable fits, whereas during
intermediate intervals the model yields unacceptable low likelihood ratios. After a certain
amount of time the standard model yields acceptable fits again, yet estimated
heritabilities are approaching zero. Overall, it was concluded that after a finite time
interval the estimations of heritability become close to zero. This result may seem
surprising, while the phenotypic time series were generated by means of the discrete
logistic equation in which parameter k and starting value x0 were under complete, or
near complete control, meaning the initially present genetic structure has seemingly
disappeared.
The authors propose an estimation scheme in which the genetic structure in
chaotic developmental processes can be recovered, and thereupon heritabilities can be
estimated. After studying the details of the developmental process (i.e. taking the type of
dynamics, elapsed time, parameter values, genetic structure etc. in consideration), time
series for each single-subject should be analyzed, yielding parameter estimates for each
subject. Subsequently, these individual parameter estimates can be analyzed under the
standard behavior genetic model. In case of their twin simulation, this was accomplished
20
by Kalman-filtering, a technique for time series analysis, yielding individual estimates
for k . Heritability of the estimated k was then estimated 1.0, and the likelihood ratio
was found acceptable. Molenaar & Raijmakers (1999) further point out that in EEG
spectra-analyses (the earlier discussed example given by Eaves et al., 1999) a procedure
resembling the introduced estimation scheme is followed.
The general conclusion drawn from the results of Eaves et al (1999) and Molenaar
& Raijmakers (1999) would be that, when in fact non-linear chaotic processes could
explain their phenotypic data, the application of a standard linear behavior genetic model
may tempt researchers to unnecessarily seek for specific environmental and/or genetic
effects, or, maybe even worse, may lead them to unwarranted conclusions concerning
these data.
21
Conclusions
In order to resolve the nature of interindividual differences, a potential third source of
phenotypic differences in development, in addition to genetic and environmental factors
has to be recognized explicitly. This independent source may consist of non-linear
epigenetic processes, which can be considered to be of chaotic origin and is the result of
autonomous self-organizing processes.
The use of the discrete logistic equation as a chaotic model for development has
showed its importance by alerting behavior geneticists to the fact that they might allocate
phenotypic differences in terms of (short-term) changes in the environment or by
(occasion specific) genetic effects. Therefore, recognizing the third source of variation
does not only have consequences for the quantitative genetic analysis of research data, as
argued below, but would also save science large amounts of time and money, which in
turn can be put in more fruitful research.
For quantitative behavior genetics the existence of a non-linear third source of
developmental differences has far reaching consequences. The straightforward traditional
model
P = G + E1 + E2 + (G × E )
appears no longer adequate, while it incorrectly subsumes non-linear epigenetic effects
under environmental influences, and should therefore be replaced with a model that
incorporates an additional term. A proposal for the latter would be EP , which denotes
Epigenesis, Epigenotype or Epigenetic influences. Hence the model including interaction
effects can be formulated as
P = G + E1 + E2 + EP + (G × E ) + (G × EP) + ( E × EP)
Perhaps in most (longitudinal) analysis individual parameters, applying advanced
mathematical techniques (e.g. Kalman filtering), have to be estimated first. Subsequently,
these estimates can be subjected to the just proposed model for further analysis.
Despite the major theoretical and practical importance and its implications for
geneticists, the discrete logistic equation as a model for development suffers from a
serious deficiency concerning psychology. The model is considerable biologically
implausible. First of all, one can argue that a discrete function is a relative poor
22
description of (human) development if it is considered a real dynamical system.
Dynamical processes unfold continuously in real time and not in steps as modeled by the
discrete logistic equation.
Secondly, this equation is a highly abstract model of development. The equation
only represents “a current state dependent on the previous state”, but what biological or
psychological state it denotes remains arbitrary. A more biological or psychological
plausible model for development and which is able to create chaotic time series would be
of additional merit. Furthermore, a model capable of self-organization would be highly
interesting in view of the discussion of epigenetics. A good example of such a model, is
the model of van Ooyen, van Pelt & Corner (1995) in which initially disconnected
neurons organize themselves into a neuronal network under the influence of their intrinsic
activity. Simulation studies in line with the design of Molenaar & Raijmakers (1999) are
recommended. Their results should be analyzed with the proposed behavior genetic
model (maybe by using individual parameter estimates).
If these data fit well to the behavior genetic model, the implications are farreaching. It would mean that the empirical evidence of a third source of variation is
supported by theory and simulations; non-linear epigenetic processes are indeed a third
source of phenotypic variability and that deterministic chaos does play a role (and maybe
a highly important one). It would mean that the so-called intangible variation can be
(partly) tangled after all and that estimated heritabilities from previous research have to
be reexamined. Finally, it would mean that the traditional nature-nurture debate has to be
broadened to a nature-nurture-epigenetics debate. In my opinion, epigenetics, selforganization and chaos theory calls for a multi-disciplinary approach (a comparison with
many components), resulting to an organized theory (a self-organizing of science) of the
human being in his environment.
23
5 References
Archer, G.S., Dindot, S., Friend, T.H., Walker, S., Zaunbrecher, G., Lawhorn, B., &
Piedrahita, J.A. (2003). Hierarchical Phenotypic and Epigenetic Variation in Cloned
Swine. Biology of Reproduction, 69, 430–436.
Barton, S. (1994). Chaos, self-organization, and psychology. American Psychologist, 49,
5-14.
Bechtel W., & Abrahamsen, A. (2002). Connectionism and the mind: parallel
processing, dynamics, and evolution in networks. Blackwell, Oxford
Beer, R. D. (2000). Dynamical approaches to cognitive science. Trends in cognitive
sciences, 4(3), 91-99.
Benno, R. H. (1990). Development of the nervous system: Genetics, epigenetics and
phylogenetics. In Hahn, M. E., Hewitt, J. K., Henderson, N. D., and Benno, R. H. (eds.),
Developmental Behavior Genetics: Neural, Biometrical, and Evolutionary Approaches,
Oxford University Press, New York. Pp. 113-143.
Bouchard, T.J. & McGue, M. (2003). Genetic and environmental influences on human
psychological differences. Journal of Neurobiology, 54(1), 4-45.
Bonabeau, E., Theraulaz, G., Deneubourg, J. L., Aron, S., & Camazine, S. (1997). Selforganization in social insects. Trends in Ecology and Evolution ,12, 188-193.
Camazine, S.. (2001). Self-organization in biological systems. Princeton University Press,
Princeton, NJ.
Eaves, L. J., Kirk, K. M., Martin, N. G., & Russell, R. J. (1999). Some implications of
chaos theory for the genetic analysis of human development and variation. Twin
Research, 2(1), 43-48
24
Eckert, M. A., Leonard, C. M., Molloy, E. A., Blumenthal, J., Zijdenbos, A. &, Giedd, J.
N. (2002). The epigenesis of planum temporale asymmetry in twins. Cerebral Cortex,.
12, 749-755.
Finch, C. E., Kirkwood, T. B. L. (2000). Chance, Development, and Aging. Oxford
University Press, New York.
Fraga, M. F., Ballestar, E., Paz, M. F., Ropero, S., Setien, F., Ballestar, M. L., HeineSuñer, D., Cigudosa, J. C., Urioste, ., Benitez, J., Boix-Chornet, M., Sanchez-Aguilera,
A., Ling, C., Carlsson, E., Poulsen, P., Vaag, a., Stephan, Z., Spector, T., Wu, Y-Z, Plass,
C., & Esteller, M. (2005). Epigenetic differences arise during the lifetime of monozygotic
twins. Proceedings of the National Academy of Sciences, 102 (30), 10604-10609.
Gärtner, K. (1990). A third component causing random variability beside environment
and genotype. A reason for the limited success of a 30 year long effort to standardize
laboratory animals? Laboratory animals, 24 (1), 71-77.
van Gelder, T. and Port, R. (1995). "It's about time: an overview of the dynamical
approach to cognition." In van Gelder, T. & Port, R. (Eds.). Mind as motion: explorations
in the dynamics of cognition. (pp. 1-43). Cambridge, MA: MIT Press
Gray, P. (2002). Psychology (4th ed.). New York: Worth.
Grigorenko, E. L. (1999). Heredity versus Environment as the Basis of Cognitive Ability.
In Sternberg, R. J. (Ed.). The Nature of Cognition. (pp. 665-696). Cambridge, MA.: MIT
Press.
Haig, D. (2004) The (dual) origin of epigenetics. Cold Spring Harbor Symposia on
Quantitative Biology, 69, 1-4
25
Jablonka, E. & Lamb, M. J. (2002). The Changing Concept of Epigenetics. Annals of the
New York Academy of Sciences, 981, 82-96.
Kowner R. (2001). Psychological perspective on human developmental stability and
fluctuating asymmetry: Applications, sources and implications. British Journal of
Psychology, 92, 447-469
Lajus, D. L., & Alekseev, V. R. (2004). Phenotypic variation and developmental
instability of life-history traits: a theory and a case study on within-population variation
of resting eggs formation in Daphnia. Journal of Limnology, 63 (Suppl. 1), 37, 44
Meinhardt, H. (1982). Models of biological pattern formation. Academic Press, London
Molenaar, P. C. M., & Raijmakers, M. E. J. (1999). Additional aspects of third source
variation for the genetic analysis of human development and behaviour: a commentary on
Eaves et al. Twin Research, 2(1), 49-52.
Molenaar, P. C. M., Boomsma, D. I., & Dolan, C. V. (1993). A third source of
developmental differences. Behavior Genetics, 23, 519-524.
van Ooyen, A., van Pelt, J., & Corner, M. A. (1995). Implications of activity dependent
neurite outgrowth for neuronal morphology and network development. Journal of
Theoretical Biology, 172, 63-82.
Plomin, R., Defries, J.C., McClearn, G.E., McGuffin, P. (2000). Behavioral Genetics, 4th
Edition. W. H. Freeman & Co
Rutter, M., Dunn, J., Plomin, R., Simonoff, E., Pickles, A., Maughan, B., Ormel, J.,
Meyer, J., & Eaves, L. (1997). Integrating nature and nurture: Implications of personenvironment correlations and interactions for developmental psychopathology.
Development and Psychopathology, 9, 335-364
26
Saunders, P.T. (1992). The organism as a dynamical system. In Stein, W. & Varela, F. J.
(Eds.). Thinking About Biology , pp.41-63, Addison-Wesley, Reading, Mass.
Smith, D. D. (1993). Brain, environment, heredity, and personality. Psychological
Reports, 72(1), 3-13.
Steinmetz, H. (1996). Structure, function and cerebral asymmetry - in vivo morphometry
of the planum temporale. Neuroscience and Biobehavioral Reviews,. 20, 587-591.
Van Speybroeck, L. (2002). From epigenesis to epigenetics: the case of C. H.
Waddington. Annals of the New York Academy of Sciences, 981, 61-81
Stige, L. C., Slagsvold, T., Vøllestad, L. A. (2005). Individual fluctuating asymmetry in
pied flycatchers (Ficedula hypoleuca) persists across moults, but is not heritable and not
related to fitness. Evolutionary Ecology Research, 7, 381-406
Thoma, R. J., Yeo, R. A., Gangestad, S. W., Lewine. J., & Davis, J. (2002). Fluctuating
asymmetry and the human brain. Laterality, 7, 45-58.
Townsend, G. C., Richards, L., Hughes, T., Pinkerton, S., & Schwerdt, W. (2005).
Epigenetic influences may explain dental differences in monozygotic twin pairs.
Australian Dental Journal, 50(2), 95-100.
Warren, K., Franklin, C., & Streeter, C. L. (1998). New directions in systems theory:
chaos and complexity. Social Work, 43, 357-372.
Wright, I. C., Sham, P., Murray, R. M., Weinberger, D. R., & Bullmore, E. T. (2002).
Genetic contributions to regional variability in human brain structure: methods and
preliminary results. Neurolmage, 17, 256-271.
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