Content uploaded by Rob G. Sacco
Author content
All content in this area was uploaded by Rob G. Sacco on Feb 12, 2023
Content may be subject to copyright.
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
Available online 18 October 2022
0079-6107/© 2022 Elsevier Ltd. All rights reserved.
Systems biology of human aging: A Fibonacci time series model
ARTICLE INFO
Keywords
Systems biology
Fractal
Fibonacci
Golden ratio
Energy ow
Self-organization
Developmental biology
ABSTRACT
Fractals are everywhere in nature, particularly at the interfaces where matter or energy must be transferred,
since they maximize surface area while minimizing energy losses. Temporal fractals have been well studied at
micro scales in human biology, but have received comparatively little attention at broader macro scales. In this
paper, we describe a fractal time series model of human aging from a systems biology perspective. This model
examines how intrinsic aging rates are shaped by entropy and Fibonacci fractal dynamics, with implications for
the emergence of key life cycle traits. This proposition is supported by research ndings. The nding of an
intrinsic aging rate rooted in Fibonacci fractal dynamics represents a new predictive paradigm in evolutionary
biology.
1. Introduction
Why do we experience physical deterioration as we age? What fac-
tors lead us to live as long as we do? Humans have been asking these
questions for thousands of years. However, it is only recently that re-
searchers have recognized the need for a systems biology approach.
Research in biology and medicine have been characterized, for the most
part, by a top-down or reductionist approach. Reductive methods are
based on the principle that biological systems are best understood
through their components. The problem of multiple levels of observation
has generally not been explicitly addressed, thus ignoring the possibility
of considering biological entities as “systems.” Current research and
theory emphasize the importance of using systems biology approaches
to studying human aging (Kitano, 2002; Torday, 2015). By combining
systems biology and reductionism, knowledge can be increased and
predictions can be improved.
Given the theoretical, empirical, and practical importance of un-
derstanding human aging, the present investigation undertakes to
describe the cause and mechanism of human aging from a systems
biology approach. With a focus on systems-level cell-cell communica-
tion, the mechanisms underlying human aging can be seen simulta-
neously in their historic and contemporary contexts. We seek to describe
how biological aging occurs in accordance with the same physical laws
that cause all matter in the universe to decay over time, the Laws of
Thermodynamics, and how genetic pathways have been evolutionarily
selected to regulate the rate of aging based on Fibonacci dynamics
(Sacco, 2013). We empirically evaluate a series of hypotheses that
pertain to the model. These hypotheses concern distinct event-related
changes of human aging, and are evaluated independently.
The present article is organized as follows. First, it denes biological
aging and provides a brief overview of the systems biology approach to
aging. Second, the law of entropy and Fibonacci dynamics are used to
explain the cause and mechanism of the aging process, identifying im-
plicit assumptions that can be empirically tested. Third, it reports the
results of an empirical review of event-related changes predicted by the
model. Finally, based on the results of the empirical review, it concludes
with overall comments regarding the ndings.
1.1. What is biological aging?
Although there is considerable dissent in the literature over how to
dene and measure aging, aging can be dened as the programmed loss
of cell-cell signaling at the whole system level caused by the loss of
energy over the passage of time and one’s interaction with the envi-
ronment (Torday, 2019a). Aging increases the probability of death.
Because biological aging includes specic event-related stages, re-
searchers often discuss biological aging in terms of development,
maturity, and senescence to make comparisons of changes across the life
span. In addition, researchers distinguish between the intrinsic and the
extrinsic rate of aging. The intrinsic rate of aging is the rate of biological
aging attributable to the genotype, and is not affected by external in-
uences (e.g., epigenetics). The extrinsic rate of aging is the rate of aging
in a population due to environmental inuences often associated with
epigenetics.
There are two main and opposite theories concerning the origin and
inevitability of natural senescence (Kirkwood and Cremer, 1982). The
stochastic aging theories posit that there is no built-in program of senes-
cence; there is only a program of development. After its end, the mature
organism could be self-maintained for a limitless time. But the repairing
efciency (e.g., damages from free radicals) affect the organism to such
a degree that its functions deteriorate to the point of no longer being
compatible with life. In programmed aging theories senescence is
conceptualized as genetically controlled. Similar to the genetic program
of development from a zygote to a mature organism, senescence is
likewise programmed (e.g., via loss of cellular signaling for homeostasis)
to facilitate new generations, which is necessary for survival of a pop-
ulation. As will become evident, this paper concerns a programmed
aging approach (Torday, 2019a).
Biological aging is often misunderstood due to the misuse of terms
such as cause and mechanism. These two terms are often used
Contents lists available at ScienceDirect
Progress in Biophysics and Molecular Biology
journal homepage: www.elsevier.com/locate/pbiomolbio
https://doi.org/10.1016/j.pbiomolbio.2022.10.005
Received 4 February 2022; Received in revised form 14 September 2022; Accepted 13 October 2022
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
25
interchangeably, but they have different meanings in the context of
aging. In order to improve clarity and precision, this paper distinguishes
between the cause and mechanism of aging. The cause of aging is a result
of the continuous action of thermodynamic forces on the organism over
its lifetime. Aging has only one cause: increasing entropy. Entropy is a
measure of disorder in a system. The entropic inuence on an organism
can be measured by the time-dependent changes in the structure and
function of molecules. Changes in molecular properties that result in the
loss of function are the mechanisms of aging. There are various mech-
anisms that can be observed at different levels and time scales, such as
direct changes in molecular structure and changes in gene expression in
the system.
1.2. Systems biology
Systems biology is a new scientic paradigm that aims at system-
level understanding of biological systems. Systems biology can be
dened as a global analysis of complex system interconnections and
their functional interrelationships (Ehrenberg et al., 2003; Kitano, 2002;
Weston and Hood, 2004). The emergence of modern systems biology has
brought together expertise from a diverse range of disciplines: biology,
engineering, mathematics, computer science, and physics. Systems
biology utilizes this expertise to determine the specic roles of the many
different components that we nd in living organisms, how these com-
ponents interact with each other, and how they all coordinate in time.
This viewpoint recognizes that multiple levels of organization encom-
passing the atomic, molecular, cellular, organismal, ecological, and
evolutionary are not independent decoupled systems and that it is vital
to provide a systematic framework for understanding the underlying
science connecting these levels.
The ultimate goal of systems biology is to quantitatively model an
organism and generate hypotheses and predictions. Integral to this is the
search for underlying causal mechanisms and principles that can be
quantied and developed into a predictive theoretical framework (West
and Brown, 2005; West et al., 2002). Systems biology identies in-
teractions among components of a system through the analysis of bio-
logical networks. Biological networks are a collection of individual parts
forming subunits or networks that are, in turn, connected to other sub-
units making up the whole. The network connections can be described.
and the properties of the network evaluated with mathematics. Systems
biology uses mathematical modeling that can ultimately lead to accurate
predictions of biological processes. In order to achieve this goal, the
interactions among critical components of a system must be identied
rst. Their functions must be determined and interconnections among
them dened.
Fractals are one of the most important features of biological net-
works. Fractals indicate the presence of self-similarity and scale-
invariance, namely that structures repeat themselves over multiple
levels of magnication or scales of measurement. Fractal-like structure
is observed in countless biological systems. This includes temporal
fractals (e.g., heart and EEG dynamics) and spatial fractals (e.g.,
vasculature and neuronal networks), which all demonstrate similar
patterns of structure (i.e., self-similarity) across multiple orders of
magnitude of time and space (i.e., scale-invariance) within the limits
and conditions of the system. This scaling or fractal structure provides a
more stable, more “error tolerant” structure that maximizes energy ow
in a system (West, 1990). As the same pattern of uctuation or
branching “runs through” a fractal, the whole structure may be char-
acterized mathematically by a single number or dimension.
1.3. Measuring biological aging
The Fibonacci Life Chart Method (FLCM) (Sacco, 2013) is a systems
biology model that integrates the 24-h circadian rhythm with the
Fibonacci sequence in order to predict intrinsic aging rates and address
complexity at a whole system level. Specically, the FLCM contends that
human aging is intrinsically rooted in the genome and expressed as
fractal time patterns. FLCM’s 24-h iterative algorithm classies
event-related stages into nine categories, namely (a) Infancy (0–2 years),
(b) Toddlerhood (2–4 years), (c) Early Childhood (4–7 years), (d) Middle
Childhood (7–11 years), (e) Adolescence (11–18 years), (f) Young
Adulthood (18–29 years), (g) Middle Adulthood (29–48 years), (h)
Older Adulthood (48–78 years), and (i) Old Age (78–127) (see Table 1).
These “stable” life stages are referred to as attractors. The life stage
patterns increase instability and threaten the stability of a prior attractor
(Sacco, 2013).
One important point implicit in the above, is that FLCM ages have a
self-organized fractal structure. These results assume importance for the
fact that the fractal organization of biological rhythms is thought to
correspond to an ideal homeostatic regulation (Torday, 2015, 2016).
Using systems biology methods, a multiscale understanding of homeo-
static regulation can be formed, expanding from the molecular level to
the whole organism. Based on the systems understanding, Fibonacci
time-series integrate various temporal scales having an essential role in
rhythmic gene expression and homeostatic regulation. Such insights
contribute to principles of cell-cell communication in order to predict
cellular behavior at the population level.
Cell-cell communication in this model is the link between genes and
epigenetics, without which there would be no “memory” upon which
evolution could depend (Torday and Rehan, 2011). In short, the Fibo-
nacci sequence is interrelated with the coevolution of genes and envi-
ronment through cell-to-cell communication, a reection of how our
physiology evolved via symbiogenesis (Sagan, 1967). Thus, Fibonacci
dynamics link the micro and macro scales. Indeed, these dynamics
explain the micro-macro relations between human developmental stages
and planetary orbital periods (Sacco, 2019a). Quantitatively under-
standing human aging at macro time scales is a new direction in systems
biology, and many questions remain to be addressed. Next, we will
examine how Fibonacci dynamics interact with entropy.
2. A systems biology perspective on human aging
Understanding human aging remains a fundamental challenge for
researchers, despite signicant effort. The systems approach is useful for
understanding the causes and mechanisms of aging, because human
biology represents a complex system consisting of a hierarchy of coop-
erating subsystems. The main idea proposed here is very simple. Since
the functioning of subsystems is subordinate to general purposes, the
Table 1
The Fibonacci life chart method.
Fibonacci Number Chronological Age Stage Transition
0 0.00
1 0.00
1 0.01
2 0.01
3 0.02
5 0.03
8 0.05
13 0.09
21 0.15
34 0.24
55 0.39
89 0.64
144 1.03
233 1.67 Infancy
377 2.70 Toddlerhood
610 4.37 Early Childhood
987 7.08 Middle Childhood
1597 11.45 Adolescence
2584 18.53 Early Adulthood
4181 29.99 Middle Adulthood
6765 48.52 Older Adulthood
10,946 78.51 Old Age
17,711 127.09 Maximum Lifespan
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
26
underlying cause of aging is ultimately founded on the physical principle
of entropy. Furthermore, the most efcient mechanism for communi-
cation between subsystems is Fibonacci dynamics. For this reason, cell-
cell signaling functions holistically, fractally, and recursively over time.
This proposition has predictable empirical effects on aging.
2.1. Entropy and the life process
The Second Law of Thermodynamics is one of the most important
principles of physics. According to the Second Law of Thermodynamics,
heat in an isolated thermodynamic system is transferred from hotter
particles to cooler ones until all the particles in the system reach a
temperature that is more or less the same (Wehrl, 1978). The universe
(as a closed system) is gradually moving toward disorder, and entropy is
increasing. However, in an open system, in which energy and work are
inputs, the distribution of energy may be affected by external forces and
feedback loops. Thus, entropy can be reduced, or even made negative
(termed negentropy) in the sense of displacement, as any decrease in
entropy will lead to an increase in entropy elsewhere (Prigogine and
Stengers, 1984).
The Second Law of Thermodynamics is the law of increasing entropy
of the universe, which constantly increases towards a maximum. To put
it differently, the Second Law is viewed as an evolutionary law,
embedded in the so-called arrow of time (Eddington, 1929; Hawking,
1988). The First Law of Thermodynamics, on the other hand, states that
the energy of the universe remains constant, and is a conservation law
allowing for reversibility in the assessment of isolated systems. Entropy
is one of the few quantities in the physical sciences that requires a
particular direction for time, often referred to as an arrow of time. The
Second Law of Thermodynamics says that the entropy of an isolated
system can increase, but not decrease as time passes. Thus, entropy
measurement distinguishes the past from the future.
A more recent hypothesis has suggested that there is a Singularity of
Nature (Torday, 2019b; Torday and Miller, 2018) through which ther-
modynamic principles can be used to reconcile physics and biology. The
origin of life can be attributed to the negentropic status of the rst
protocells as a result of the equal and opposite reaction to the Big-Bang
according to Newton’s Third Law of Motion. The enduring thermody-
namic coupling between the outward environment and the interior of
the cell allowed the rst primitive life to exist. Torday (2016) suggests
that the Earth’s protocells originated from lipid-based micelles that
could form and deform in response to the sun’s heat while maintaining
structural integrity. Hysteresis is the source of molecular memory,
which is vital for evolutionary processes (Torday and Rehan, 2011).
The cell became the rst self-organizing process as a result of inter-
action between the external and internal environments (Torday, 2015).
The evolutionary leaps that followed, such as endosymbiosis were re-
iterations of this self-organizing process. Therefore, evolution can be
understood as the continuous internalization of the environment gov-
erned by the thermodynamic laws. Additional evolutionary develop-
ment represents the multicellular augmentation of these fundamentals,
based on principles of cell-cell communication, which extend
throughout embryologic development. In this way, multicellular coop-
eration, human physiology, society, and culture all refer to the
Singularity/Big-Bang as fractal iterations.
The framework aligns with the nilpotent universal computational
system (Marcer and Rowlands, 2017). The biological reproduction re-
iterations that always refer to the origin of the Singularity are a creation
operation that “conserves and proofreads” to ensure life’s integrity. In
the context of quantum theory, a zero-point origin can be linked to the
Fibonacci numbers. The Fibonacci numbers wind fractally around the
zero-point, while constantly receding from it. The spiral represents the
inception, progress, and completion of various stages of development
(Sacco, 2013). Consequently, as evolution progresses, it always begins at
the completion of a prior stage (which corresponds to the focus of an
internal zero-point), expanding outward from the center, and thus
representing evolution.
2.2. Entropy as the cause of aging
Maintaining organization through cell-to-cell communication re-
quires energy expenditure. The majority of the consumed energy is spent
during the fertile reproductive phase of life such that maximum
negentropic organization processes reach their peak during this time
(Torday, 2019a). Once the reproductive years are complete, the other-
wise optimal bioenergetic state of the organism begins to break down
since the amount of bioenergy available over the course of the life cycle
is nite. This is what is described as the aging process (Torday, 2019a).
Given this, aging is predicated on an intrinsic rate of aging for the loss of
cellular signaling for homeostasis. This step-wise retrograde dissolution
of physiology is commonly referred to as “healthy aging,” whereas
various diseases that can either undermine and/or accelerate the innate
aging process are referred to as disease-based aging.
There is strong evidence that the human lifespan responds to
entropic changes. Lifespan entropy generation is affected by diet (Silva
and Annamalai, 2009), activity level (Silva and Annamalai, 2008), heat
loss (Aoki, 1994), and the size of the body (Demetrius et al., 2009). Silva
and Annamalai (2008) used a thermodynamic denition of entropy to
calculate the average lifespan of humans. It was found that entropy
generated over the lifespan predicts a lifespan of 73.78 years and 81.61
years for average males and females in the United States, respectively,
which is very similar to the statistical average for both sexes of 78.6
years (Centers for Disease Control and Prevention, 2017). Researchers
have also examined the entropy generated by a human at different ages.
According to Aoki (1994), human entropy production increases rapidly
from birth to age 18, then slowly declines thereafter. This supports a key
point in the cell-to-cell communication theory of aging, which is that
energy dissipation has a bimodal distribution and peaks during the
reproductive years (Torday, 2019a).
2.3. Fibonacci dynamics as a mechanism for maximizing entropy
Fibonacci dynamics generates a sequence of numbers starting from
the states 0 and 1. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, etc.) is
the simplest recursive number sequence possible whose terms are the
sum of the previous two terms. The Fibonacci sequence and Golden
Ratio (about 1.618034) are complementary in the sense that the ratios of
the successive Fibonacci numbers converge on the Golden Ratio (Livio,
2008). Throughout nature, the Fibonacci sequence and Golden Ratio can
be found. About 90% of all owering plants display seed or leaf patterns
based on Fibonacci numbers (Jean, 1994). DNA molecules also follow
this pattern, measuring 34 Å long and 21 Å wide for each cycle of the
double helix (Perez, 2010; Yamagishi and Shimabukuro, 2007). The
Fibonacci sequence and Golden Ratio also appear in human anatomy as
body proportions (Ferring and Pancherz, 2008), bronchial airway
segment bifurcations (Goldberger et al., 1985), and hair whorls (Paul,
2016), among others. Research has also highlighted a range of biological
rhythms that are under the control of the Golden Ratio.
For example, the clock cycle of brain waves is linked to the Golden
Ratio (Roopun, 2008; Weiss and Weiss, 2003). Body temperature regu-
lation is another example of the Golden Ratio in action. Body temper-
ature is a classic phase marker for the circadian rhythm. During sleep,
body temperature falls to a minimum at around 04:00 hours, and rises
until 18:00 hours (Minors and Waterhouse, 1981). On a 24-h clock, such
periods of minimum and maximum ux occur close to the hours 04:00
and 19:00, which are at an angle of 137.5◦(the Golden Angle). In this
regard, body temperature uctuates within the range of the Golden
Ratio, with exogenous sources of variability, such as sleep and exercise
(Hiddinga and Van Den Hoofdakker, 1997). Furthermore, the Golden
Ratio is also relevant to understanding the circadian rhythm of the
cardiovascular system. In their study, Yetkin et al. (2014) found that the
24-h systolic to diastolic and diastolic to pulse pressures are in close
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
27
agreement with the Golden Ratio. Based on these diverse examples, it is
reasonable to conclude that the Golden Ratio is an important factor in
human biology.
Why are the Fibonacci numbers and Golden Ratio so important in
human biology? One simple answer would be the entropy law. Animate
and inanimate forms have characteristics expressible as Fibonacci
numbers or the Golden Ratio because this self-similar fractal dynamic
reects global efciency with a maximum entropy constraint (Bejan and
Zane, 2013). The harmonic orbital resonances of the planets in the solar
system are also reective of Fibonacci dynamics because they facilitate
entropy export (Sacco, 2019a). The spiral patterns found in galaxies
(Livio, 2008), hair whorls (Paul, 2016), and sunowers (Jean, 1994)
reference Fibonacci numbers and/or the Golden Ratio, minimizing ow
resistance. Another pattern in nature that maximizes ow is tree-like
branching. As seen in river drainage (Sharp, 1971), trees (Jean, 1994),
and bronchial airway segments in the lung (Goldberger et al., 1985),
they are reective of Fibonacci numbers and/or the Golden Ratio. Life
keeps its entropy low by releasing entropy into the environment through
breathing and metabolism. The term negentropy refers to keeping en-
tropy low (Schr¨
odinger, 1967). Essentially, Fibonacci numbers and/or
the Golden Ratio reect the most negentropic/entropic organization
processes and minimum ow resistance in most systems.
Given that the Fibonacci sequence and Golden Ratio reect crucial
roles in human biology, research has investigated the signicance of the
Fibonacci sequence in human development (Sacco, 2013). To capture
the various event-related stages of aging at a systems level, the Fibonacci
Life Chart Method (FLCM) describes a series of nine life stages, from
infancy to old age (see Table 1). As entropy increases in the body, its
efciency declines (e.g., as cells die or become disordered), and the
body’s ability to produce the same output at the same energy input
decreases. Therefore, in order to decrease entropy, the efciency of the
body must be increased. Essentially, this means event-related reorga-
nization based on the Fibonacci sequence. Life stages, therefore, have as
their purpose the reorganization of cell-cell communication to enhance
energy efciency. This process occurs fractally throughout the life span
because fractals minimize the energy lost during communication within
the cell-cell network (see Fig. 1).
A simple model will facilitate the discussion of these issues (Fig. 2).
The relevant variables are entropy, efciency of cell-cell communica-
tion, and the Golden Ratio. If 100% absolute efciency could be
attained, cell-cell communication would have no entropy and would
achieve 100% homeostasis with respect to the Golden Ratio. Obviously,
this is not possible, but efciency can be increased gradually to minimize
entropy growth by approximating the Golden Ratio, as shown by the
convergence of Fibonacci numbers towards the Golden Ratio. On the
other end, a condition of zero efciency results in maximum entropy.
This is a state where disorder has increased to a point where cell-cell
communication is incapable of controlling the signaling and function
between cells to maintain microenvironmental homeostasis, and as a
result life processes cease. We can now turn to a discussion of the
empirical predictions of this model.
2.4. Empirical predictions
The hypothesis that the distribution of bioenergy over the course of
the life cycle is skewed towards Fibonacci dynamics permits empirical
predictions. A rst basic prediction is that the intrinsic rate of aging
represents a specic strategy that enables a set of systems to acquire
traits that improve negentropy as entropy in the body increases. That is,
the amount of energy gained by a trait per amount of energy consumed
to generate that gain should be maximized. An example would be
muscles and organs providing more output with less effort. For the brain,
this involves reducing its effort to learn new things, to solve problems,
and to create new ideas. For the peripheral nervous system, it involves
increased synaptic transmission. Such patterns are plausible with other
event-related changes in development, and such evidence will be
considered in more detail later.
A second prediction is that the intrinsic rate of aging will have an
optimal energy distribution that corresponds with Fibonacci dynamics.
If human aging is malleable due to phases of continuity and change, then
discontinuous processes will be reected by Fibonacci dynamics. Simply
Fig. 1. The Fibonacci spiral fractal grows outward by a factor of the Golden
Ratio for every 90 degrees of rotation. It is approximated using quarter-circle
arcs inscribed in squares with sides the length of the Fibonacci numbers.
Fig. 2. The inverse relationship between entropy and efciency. The solid
line shows the efciency limit, which is asymptotically reached as Fibonacci
numbers approximate the Golden Ratio. The efciency limit is, of course, not
reached. Life resists maximum entropy, and maximum entropy is death.
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
28
put, discontinuity depends on fractal time scales to integrate the external
and internal contexts with maximal efciency, and as a result, fractal
patterns will correlate with human aging. Methodologically, this pre-
diction is a useful complement to the rst because it is aligned with the
claim that human aging is simply a strategy to maximize energy ef-
ciency and homeostasis. The energy efciency hypothesis predicts that
developmental change will show high correlations with Fibonacci dy-
namics, while developmental continuity will show lower correlations
with Fibonacci dynamics.
A nal prediction is that epigenetic effects (i.e., extrinsic rate of
aging) may inuence the regulation of gene expression leading to phe-
notypes (i.e., intrinsic rate of aging). Biological aging is highly indi-
vidualized because of differences in interactions with the environment.
For example, the timing of puberty is affected by many factors, including
genetic and biological inuences, stress, socioeconomic status, health,
and environmental factors. The intrinsic rate of aging reects an in-
dividual’s genotype, and it can be measured accurately. It is more
challenging to measure the extrinsic rate of aging, which is the result of
the environment and the genotype interacting. That is, like the atom, the
cell is both deterministic and probabilistic (Torday, 2018). Therefore,
the Fibonacci time series model is only valid for predicting human
developmental outcomes when aggregated. Statistics should be
compared on the basis of population averages. The average statistic of a
population, which is an average of all individuals, is assumed to be
approximately the same as that of the deterministic model.
2.5. Fibonacci dynamics and human aging: evidence for a correlation
Having postulated that Fibonacci time series regulates the negen-
tropy maximizing cell-cell signaling interactions relevant to the intrinsic
rate of aging, we turn our attention to research evidence relevant to the
theory. We examine empirical evidence that Fibonacci dynamics are
related to seven changes in human aging: (1) prenatal growth, (2) motor
control, (3) perception, (4) brain development, (5) muscle and bone
strength, (6) reproductive changes, and (7) senescence. We discuss each
of these lines of evidence in turn.
2.6. Prenatal growth
Conception involves fertilization of an egg cell with a sperm cell. The
period from conception to birth (termed prenatal development) is a
period of rapid growth, cellular differentiation, and organ system
maturation. From a single zygote, 2 cells will develop, then 4, 8, 16, and
so on until about 200 billion cells present at birth. Fibonacci dynamics
correlate to all of the prenatal milestones.
1. Fertilization
After fertilization, human embryos double every 24 hours (Lagar-
kova et al., 2010), based on biological periodicity stored in the genomes
of all organisms. Fibonacci numbers follow the doubling sequence
mathematically (Linage et al., 2006), explaining the connection between
the Fibonacci numbers, 24-h clock cycle, and embryonic development.
2. Implantation
During implantation, the fertilized egg adheres to the uterine wall. It
takes 7 days. The sum of the rst 4 Fibonacci numbers is 7 days (1 +1 +
2 +3 =7 days). Thus, the Fibonacci model gives exactly similar results.
3. Embryonic Period
The embryonic period is when organogenesis begins. It is a period of
rapid tissue differentiation. The embryonic period is complete after 56
days of pregnancy (Klieger, 2013). At this stage, the embryo is known as
a fetus and is approximately 1 inch long. Note that 54 days equals the
sum of the rst 8 Fibonacci numbers (1 +1 +2 +3 +5 +8 +13 +21 =
54 days). Thus, the Fibonacci model gives almost similar results, with a
percentage difference of 3.6%.
4. Second Trimester
During the second trimester, the fetus undergoes a more continuous
development and growth cycle. The second trimester begins after 84
days of pregnancy (Klieger, 2013). Fibonacci numbers 1 through 9 add
up to 88 days (1 +1 +2 +3 +5 +8 +13 +21 +34 =88 days). Thus,
the Fibonacci model gives almost similar results, with a percentage
difference of 4.7%.
5. Birth
In general, a normal human pregnancy is about 40 weeks long (280
days). As per the Fibonacci sequence, 232 days is the sum of the rst 12
numbers (1 +1 +2 +3 +5 +8 +13 +21 +34 +55 +89 =232 days).
The Fibonacci model gives inconsistent results for the timing of birth,
with a percentage difference of 18.8%. Although all the major prenatal
developmental processes are complete by 232 days, the predicted model
age would correspond to a preterm birth, dened as the birth of a baby at
fewer than 37 weeks (259 days) (Brown et al., 2013). In the United
States 12.3% of all births are before 37 weeks (Mathews et al., 2011).
Most preterm infants survive with appropriate treatment. A number of
scientic geniuses were reportedly born prematurely, such as Kepler,
Newton, Darwin, and Einstein, and it has been suggested that prema-
turity and genius may be linked. However, the evidence for this is
inconclusive (Dutton et al., 2021).
2.7. Motor control
The fundamental prediction of the programmed aging model is that
the transition from one stage of life to another is highly associated with
Fibonacci dynamics. Consistent with the theory, explicit changes in
motor control appear to be correlated with Fibonacci ages. Events such
as independent walking are typically used as an indicator of the progress
of motor development in early life, and it has been found that infants
start walking at around age 1 (Størvold et al., 2013). Studies also show
that rst signs of hand dominance may be observed at age 1.5 (Loring
et al., 1989). Furthermore, children aged 7 have achieved mature bal-
ance control, indicating that the development of structures responsible
for motor control is complete (de S´
a et al., 2018).
Nelson (1983) rst suggested the idea that planning in the central
nervous system may be optimization of movement to minimize energy
consumption. In a reaching movement, for example, the optimal
movement consumes the least energy among all the movements which
accurately reach to a target. Following this idea, research has found that
the dynamic balance control during human walking can be described by
the Golden Ratio (Iosa et al., 2017). This may be because the Golden
Ratio optimizes movement to minimize energy consumption, whereas
the degree to which walking is less coordinated consumes more energy.
2.8. Perception
Human perception is designed to process multisensory information.
If perceptual sensing processing signals were unrelated to energy
expenditure in scanning for threats or available energy from food
sources, there would be no particular reason that Fibonacci dynamics
would relate to the development of the human perceptual system. In
contrast, if it is correct to conceptualize Fibonacci dynamics as an index
of programmed aging relationships at a whole systems level, develop-
mental changes related to perception should have a strong correlation
with Fibonacci ages compared to other ages, because Fibonacci ages
have greater implications for energy efciency.
The empirical data strongly support the prediction that programmed
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
29
aging exerts a stronger effect at Fibonacci ages than at other ages. For
example, it is known that at age 7, spatial contrast sensitivity and
temporal contrast sensitivity reach adult levels in visually normal chil-
dren (Ellemberg et al., 1999). Following birth, the central auditory
pathways and processing abilities continue to gradually mature and are
considered complete by age 7 for most auditory skills (Werner, 2020).
Such a pattern documents clear evidence of age-related genetic pro-
gramming based on Fibonacci dynamics. This nding is difcult to
explain if we assume that perceptual sensing is unaffected by optimi-
zation to maximize energy efciency.
2.9. Brain development
If, as we have proposed, optimal brain processing and cognition is
related to distribution of energy over the life span, we should nd strong
links between changes in brain development and cognition that corre-
late with Fibonacci ages. The literature is rife with such connections.
1. Structural Brain Changes
The brain has the most abundant energy metabolism in the human
body. Although it accounts for only 2% of total body weight, the brain
requires approximately 20% of the total body energy in the resting
awake state (Sokoloff, 1999). Glucose is the main energy substrate and
demands of the brain vary according to neuronal activity. The brain’s
functional connectivity is complex, has high energetic cost, and requires
efcient use of glucose, the brain’s main energy source. Regions with a
high degree of functional connectivity are energy efcient and can
minimize consumption of glucose (Tomasi et al., 2013). The higher
energy demands of brain communication that hinges upon higher con-
nectivity could render brain hubs more vulnerable to decits in energy
delivery or utilization and help explain their sensitivity to neurode-
generative conditions, such as Alzheimer’s disease.
The high energetic cost of human brain function can only be main-
tained through a combination of strategies for efcient energy use. For
example, due to the fact that our heads are too large to t through the
birth canal in a fully mature state, we are born with only 25% of our
adult brain size. This is a consequence of bipedalism, made possible by
endothermy, allowing the forelimbs to form tools, especially speech
(Torday, 2015). Consequently, the human central nervous system and
the evolution of consciousness have been subjected to intense positive
selection pressures (Torday, 2020), including imagination in humans
(Torday, 2021). The brain undergoes dramatic postnatal growth, so by
the age of 2 years overall brain size reaches about 85% of adult volume
(Knickmeyer et al., 2008). The rapid pace of structural brain develop-
ment is also accompanied by an equally rapid development of cognitive
and motor functions. The brain continues to undergo structural and
morphological changes throughout the various stages of development
and aging.
Around 6–8 years of age, the adrenal glands begin producing
androgen (Auchus and Rainey, 2004; Bogin, 1997; Hochberg, 2008).
Despite their minor effects on physical development, adrenal androgens
have profound effects on the functioning of the brain. Adrenal andro-
gens act as sex hormones and activate sexually differentiated brain
pathways (Del Giudice, 2009). In adolescence, the brain experiences a
unique period of maturation. A process of thickening gray matter,
arborization or synaptic density is known as “synaptogenesis.” It peaks
between 11 and 12 years old (Feinberg and Campbell, 2010; Giedd,
2004). A decline follows this peak, a process known as “pruning,” where
extra synapses are removed. Synaptic pruning is thought to be the
brain’s way of eliminating connections that are no longer needed. Gray
matter volume changes around ages 11–12 are considered to be
nonlinear and regionally specic, with the frontal lobes maturing later
in development (Giedd et al., 1999). For example, it has been found that
microstructural changes consistent with brain maturation patterns in the
prefrontal cortex increase rapidly with age until age 18, followed by a
less steep increase until age 47, after which a decline in brain changes is
observed (Falangola et al., 2008).
These assorted ndings are consistent with Fibonacci dynamics.
Brain dynamics are complex, involving several different and interacting
temporal scales, that is, they are fractal. For example, fast electric ac-
tivity, slower chemical reactions, and even slower aging processes are
observed in the brain. Importantly, these brain dynamics are “scale-free”
(Fraiman and Chialvo, 2012; Stam and de Bruin, 2004), meaning that
certain signaling properties stay preserved across different time scales.
To describe and quantify such time scale invariant dynamics, the
framework of fractal geometry is often applied (Werner, 2010).
Furthermore, it is known that the fractal dynamics of the brain are based
on the Golden Ratio (Roopun, 2008; Weiss and Weiss, 2003). The
biggest gap in the literature so far, however, is how fractal time series
relate to brain dynamics over longer time spans. The Fibonacci Life
Chart Method (FLCM), to our knowledge, is the only systems-level time
series model that can account for the organizing principles of brain
dynamics just reviewed.
2. Cognitive Development
Several studies have documented a relationship between Fibonacci
ages and optimal cognitive function. Optimal cognitive function is very
important in every stage of human development. Optimal cognitive
function refers to processes such as learning, reasoning, attention, and
decision making. The acquisition of language is a crucial part of cogni-
tive development. Most children produce their rst recognizable words
between the ages of 1 and 1.5 years (Rosselli et al., 2014). The
remarkable speed of language acquisition observed in children aged 1.5
years is attributed to neuronal changes (e.g., the growth of axons and the
number of dendrites) as well as increased myelination processes that
allow faster conduction (Rosselli et al., 2014). Studies have also shown
that by the time children are 4–5 years old they have mastered nearly all
the rules of their native language, and they are capable of producing and
comprehending most of the grammatical structures, with subsequent
development focusing primarily on increasing vocabulary (Werker and
Hensch, 2015). While development of early language skills varies
greatly among children, the evidence is consistent with our theory that
potential demands on mental energy in language production should
correlate with Fibonacci ages.
The cognitive processes that mediate self-awareness and identity are
related to Fibonacci ages 1.7 and 2.7. For example, a few studies have
addressed the question of self-recognition, an integral aspect of self-
awareness. Typically, self-recognition is assessed by the mirror task
(Gallup, 1970). The present evidence indicates that according to mirror
task standards, 1-year-olds do not possess self-awareness, 50% of
1.5-year-olds are self-aware, and by 2 years of age infants have achieved
self-awareness (Bard and Leavens, 2009; Lewis and Brooks-Gunn, 1979).
Furthermore, the development of other behaviors related to
self-awareness, such as pretend play and synchronic imitation (Nielsen
and Dissanayake, 2004), have also been observed at the 1.5-year mark.
According to research, 1.5-year-olds are at a distinct stage of develop-
ment (Bertenthal and Fischer, 1978; Nielsen and Dissanayake, 2004;
Stapel et al., 2016) and their brain scans differ from 1 or 2-year-olds. In
the same vein, by the age of 2.5, children begin to identify themselves as
either boys or girls (Volbert, 2000). As many social functions critically
rely on gender perception, gender recognition is a fundamental task for
humans.
During the 2nd year of life, long-term memory becomes increasingly
reliable as the hippocampus indexes memories (Bauer, 2002; Rubin,
2000). Children begin to recount autobiographical events around the
age of 4 (Nelson, 1992). However, their reasoning is still largely
instinctual and not entirely logical. More complex concepts, such as
cause and effect, time, and comparison, remain beyond their grasp.
Around the age of 7, children enter what Piaget called the concrete
operational period, which lasts until about the age of 11 (Piaget, 1952).
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
30
During this time, children develop a better understanding of and facility
with mental operations (e.g., can think about how to approach a prob-
lem and consider different outcomes). By age 11, children are also able
to recognize facial emotions at an adult level (Chronaki et al., 2014).
Human social interactions depend heavily on facial recognition, and the
cognitive mechanisms of facial recognition have been well studied. For
example, it has been found that the ability to recognize new faces peaks
around age 30, and then declines (Germine et al., 2011). All of this
conrms the view that Fibonacci ages support optimal cognitive
function.
There is strong support for cognitive abilities at age 18.5. Indeed,
researchers have discovered that overall brain processing power and
detail memory peak at age 18 (Hartshorne and Germine, 2015). There is
evidence that this age is an important time when people develop a sense
of purpose, question their spiritual beliefs, and formulate or conceptu-
alize what it means to live a meaningful life. One study using a large
sample of young adults found that age 18 is a predictor of spiritual
experience (Sacco, 2017). The fact that people report more spiritual
experiences at this age tells us something important about how Fibo-
nacci ages can have important applications to the science of spirituality
(Sacco, 2019b). In addition, although some personality changes may
occur in middle adulthood, overall personality stability begins around
age 30 (Terracciano et al., 2006).
2.10. Muscle and bone strength
Programmed aging theory predicts that Fibonacci dynamics are most
important to muscle and bone strength across the lifespan. Consistent
with this, Fibonacci dynamics correlates very highly with muscle and
bone strength. For example, muscle strength has been found to increase
up to age 29, remains constant to age 49, and then decreases with
increasing age (Larsson et al., 1979). Likewise, bone mass increases
during adolescence, reaches a plateau at age 30, and remains stable until
about age 50, after which bone mass gradually decreases (Hodges et al.,
2019; Stagi et al., 2013). Clearly, we observe a strong correlation be-
tween muscle and bone strength, with peaks in both tied to Fibonacci
dynamics.
Furthermore, dental development patterns are highly consistent with
Fibonacci dynamics. Indeed, the Golden Ratio value of 1.618 is
expressed in the dimensions of teeth (Anand et al., 2017). This ts with
evidence that the Golden Ratio and dental development are correlated.
For example, there is a period of “teething” lasting about 2.5 years, with
the rst tooth erupting between 6 and 13 months, and all other teeth
erupting before the age of 3 (Nelson, 2020). Children begin eating solid
foods at age 2.5 years. The function of teeth is more efcient chewing
and eating. In the context of energy intake, people who eat faster tend to
consume more energy during a meal (Robinson et al., 2014).
Except for the third molars, all permanent teeth erupt in two stages,
between the ages of approximately 6–8 years, followed by a silent
period, and then again between 10 and 12 years of age (Nelson, 2020).
Upper second molars are the last permanent teeth of the upper arch to
erupt, and erupt at a chronological age of 11–12 years (Nelson, 2020).
Third molars, or wisdom teeth, are the teeth situated the furthest from
the dental arch, and appear late in development. They usually erupt and
complete development at the age of 18. This nding is used as a human
biological growth marker demonstrating unambiguously that a subject
has attained the age of 18 years (Roberts et al., 2016).
2.11. Reproductive changes
The average girl reaches puberty at the age of 10, and the average
boy at the age of 12 (Kail, 2022). Puberty refers to changes in the process
of sexual maturation, including changes in the reproductive organs and
the development of secondary sexual characteristics such as facial and
body hair. Menarche is dened as the onset of menstruation and signies
that the body is readying for reproduction. The average age of menarche
is presently around 12.3 years in the United States, with variations by
race and ethnicity (Finer and Philbin, 2014). In developing countries,
the age at which menarche occurs has decreased by almost 4 years over
the past 150 years. However, it appears as though this trend has ended,
and the age of menarche has reached a plateau. The decrease in the age
of menarche may be due to improvement in socioeconomic conditions,
effective public health measures, and nutrition.
We suggested that a genetic mechanism that monitors bioenergy of
vital importance to the life cycle would be expected to peak at the
reproductive stage. If we compare the peak in maximum negentropic
processes at age 18 (Aoki, 1994), we easily see that this peak synchro-
nizes with the fertile reproductive phase of life. Thus, a human female’s
period of optimal fertility starts at the age of 18 years (De Bruin and te
Velde, 2004). In connection with the timing of optimal fertility, at age
21, when females are at the peak of their fertility, the ratio of a uterus’s
length to its width is 1.618, which is equal to the Golden Ratio (Verguts
et al., 2013). In males, testosterone peaks at the age of 18–19, and then
declines with age (Kelsey et al., 2014). An evolutionary perspective
suggests that the peak in testosterone reects an adaptational response
to the reproductive stage, since multiple aspects of sexual behavior are
inuenced by testosterone (Edelstein et al., 2011).
As women enter midlife, they experience a major biological process
during which they pass from their reproductive to nonreproductive
years. The loss of the natural ability to bear children is usually complete
by age 50 (Eijkemans et al., 2014). Menopause is the point at which
menstruation stops. Menopause occurs as the woman’s ovaries slow in
their production of estrogen, and eventually stop releasing eggs. The age
at natural menopause (ANM) varies from 40 to 60 years, but the global
average ANM is 48.8 years (Schoenaker et al., 2014). Several factors
inuence the timing of menopause, including genetics, lifestyle factors
(such as stress, smoking, and obesity), and reproductive status. Many
genes are epigenetically regulated during this process, including those
involved in DNA repair and immune function (Stolk et al., 2012).
To the extent that menopause can be conceptualized as an adjust-
ment to the loss of fertility and concerns about physical attractiveness
and health (Rossi, 2004), menopause raises the possibility of depression
(Freeman, 2010). During aging, men experience a decline in testos-
terone, possibly even a low-testosterone syndrome associated with
depression and anxiety (Amore et al., 2009). As a result of the biological
and social changes experienced by women and men in older adulthood,
depression is highest at age 48.5 around the world (Blanchower and
Oswald, 2008).
2.12. Senescence
Senescence is the gradual deterioration of functional characteristics
that result in death. Senescence is a highly regulated process involving
molecular, biochemical, and physiological changes. Changes in physical
features are the most obvious senescent changes. The hair becomes
thinner and grayer. Hair loss by age 50 affects about 50% of men and
25% of women (Vary, 2015). Facial wrinkles are noticeable by the age of
50 in men and women (Luebberding et al., 2014). Hearing problems are
the most frequent type of impairment reported by older adults.
Age-related hearing loss is nearly ubiquitous among older adults, with
nearly 80% of adults over age 80 years demonstrating signs of clinically
signicant hearing impairment (Bernabei et al., 2014). Neuroimaging
studies have shown that aging is also associated with atrophy of the
brain and disruption of structural and functional connections in the
brain (Hafkemeijer et al., 2014). Aging may also be associated with
dementia, a decline in mental ability that interferes with daily living.
Dementia affects 5% of the elderly over 65, and 25% of those over 85
(Archer et al., 2015). The average age of onset for dementia in males is
78.8 and in females it is 81.9 (Brinks et al., 2013).
Statistically speaking, life expectancy measures the average lifespan
of a human. Life expectancy is increasing worldwide. In the United
States, the average life expectancy at birth is 78.6 years (Centers for
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
31
Disease Control and Prevention, 2017). Maximum life expectancy is a
somewhat disputed topic, which may differ among populations, based
mainly on genetic factors. Studies indicate a theoretical maximum life-
span for humans of approximately 126 years (Weon and Je, 2008). Thus,
the research suggests that the FLCM model is a reliable predictor of both
average and maximum human life expectancy.
2.13. Summary
Overall, 42 event-related changes were identied across nine major
stage transitions included in this review. A summary of the empirical
ndings is presented in Table 2. These ndings strongly support several
hypotheses derived from systems biology of human aging. As the theory
predicts, the intrinsic rate of aging correlates with Fibonacci dynamics
and event-related stages considered at the population level. Further-
more, the acquired phenotypes resulting from the interaction of the
genotype with the environment assist in maximizing negentropy, and
the phenotypes act in ways that suggest they are adaptations to
increasing entropy. Thus, the results of this correlational review support
a link between Fibonacci dynamics on one hand, and phenotypic
expression on the other. Taken as a whole, the data consistently show
that phenotypes correlate with Fibonacci dynamics that have implica-
tions for maximizing negentropy. Clearly, phenotypic expression is tied
closely to Fibonacci dynamics and their fractal properties with reference
to the First Principles of Physiology (i.e., negentropy, chemiosmosis, and
homeostasis) (Torday and Rehan, 2009). Such effects are consistent with
a model of cell-cell communication that links Fibonacci dynamics to the
negentropic maximizing tendencies of human development, maturity,
and aging.
3. Conclusion
The cause of human aging is the process of energy dissipation. In this
paper, a systems biology of human aging was established based on en-
ergy dissipation and the mechanism of Fibonacci dynamics. Further-
more, empirical predictions of the model were examined using data on
human aging. The main conclusions are as follows: (1) genetic pathways
have been evolutionarily selected to regulate the rate of aging and ac-
quire traits that improve energy efciency and homeostasis; (2) human
aging exhibits a prominent fractal character based on Fibonacci dy-
namics, with the fractal ages being 1.7, 2.7, 4.4, 7.1, 11.5, 18.5, 30, 48.5,
78.5, and 127; and (3) epigenetic effects may inuence the regulation of
gene expression leading to phenotypes. The systems biology model
proposed in this paper can be used to predict the intrinsic rate of aging
based on the distribution of energy over the life span and is expected to
provide useful guidance for improving research and clinical decision-
making.
Researchers have puzzled over how people change (and stay the
same) for decades. The present systems biology model has made possible
a predictive approach to human aging. The model and empirical ndings
discussed in this article may nally help us understand the complexities
of human aging. The Fibonacci sequence is an example of the unifying
order of creation. Fibonacci patterns are everywhere in nature,
reminding us that the same dynamics shape galaxies, planets, whirl-
pools, sunowers, and our own DNA. The present research is useful not
only for advancing our understanding of biological processes but also as
a guide to understanding how biology, individual behavior, and the
larger environment interact. Since mathematics originates in the uni-
verse and has been assimilated over the course of evolution, the Fibo-
nacci sequence is indicative of symbiogenesis. By exploring these links,
we may be able to attain a new integrative approach to knowledge that
encompasses the entire range of human experience.
Declaration of competing interest
We have no conict of interest to disclose.
Acknowledgements
John S. Torday was a recipient of NIH Grant HL055268.
References
Amore, M., Scarlatti, F., Quarta, A.L., Tagariello, P., 2009. Partial androgen deciency,
depression and testosterone treatment in aging men. Aging Clin. Exp. Res. 21, 1–8.
https://doi.org/10.1007/bf03324891.
Anand, R., Sarode, S.C., Sarode, G.S., Patil, S., 2017. Human permanent teeth are divided
into two parts at the Cemento-enamel junction in the Divine Golden Ratio. Indian J.
Dent. Res. 28, 609. https://doi.org/10.4103/ijdr.ijdr_525_16.
Aoki, I., 1994. Entropy production in human life span: a thermodynamical measure for
aging. Age 17, 29–31. https://doi.org/10.1007/bf02435047.
Archer, H.A., Smailagic, N., John, C., Holmes, R.B., Takwoingi, Y., Coulthard, E.J.,
Cullum, S., 2015. Regional cerebral blood ow single photon emission computed
tomography for detection of frontotemporal dementia in people with suspected
dementia. Cochrane Database Syst. Rev. https://doi.org/10.1002/14651858.
cd010896.pub2.
Auchus, R.J., Rainey, W.E., 2004. Adrenarche - physiology, biochemistry and human
disease. Clin. Endocrinol. 60, 288–296. https://doi.org/10.1046/j.1365-
2265.2003.01858.x.
Bard, K.A., Leavens, D.A., 2009. Socioemotional factors in the development of joint
attention in human and ape infants. In: R¨
oska-Hardy, Louise, Neumann-Held, E.M.
(Eds.), Learning from Animals? Examining the Nature of Human Uniqueness. Psy-
chology Press, Hove, pp. 89–104.
Bauer, P.J., 2002. Long-term recall memory: behavioral and neuro–developmental
changes in the rst 2 years of life. Curr. Dir. Psychol. Sci. 11, 137–141. https://doi.
org/10.1111/1467-8721.00186.
Bejan, A., Zane, J.P., 2013. Design in Nature: How the Constructal Law Governs Evo-
lution in Biology, Physics, Technology, and Social Organizations. Anchor, New York
(NY).
Table 2
Summary of empirical ndings.
Chronological Age Stage Transition Milestones
0.00 0. Prenatal Growth 1. Fertilization
0.01 2. Implantation
0.06 3. Embryonic period
0.09 4. Second trimester
0.24 5. Birth
1.03 1. Infancy 6. First steps
7. First words
1.67 8. Hand dominance
9. Language development
10. Self-recognition
2.70 2. Toddlerhood 11. Brain growth
12. Gender recognition
13. Long-term memory formation
14. Teething completed
4.37 3. Early Childhood 15. Autobiographical memory
16. Language complexity
7.08 4. Middle Childhood 17. Androgen secretion
18. Balance control
19. First molar eruption
20. Visual perception
21. Auditory skills
11.45 5. Adolescence 22. Facial emotion recognition
23. Puberty
24. Synaptogenesis
25. Second molar eruption
18.53 6. Early Adulthood 26. Peak memory
27. Peak testosterone
28. Spiritual experience
29. Start of optimal fertility
30. Third molar eruption
29.99 7. Middle Adulthood 31. End of optimal fertility
32. Peak facial recognition
33. Peak bone mass
34. Peak muscle strength
35. Personality stability
48.52 8. Older Adulthood 36. Facial wrinkles
37. Hair loss
38. Menopause
39. Peak depression
78.51 9. Old Age 40. Average life expectancy
41. Dementia onset
127.09 42. Maximum life expectancy
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
32
Bernabei, R., Bonuccelli, U., Maggi, S., Marengoni, A., Martini, A., Memo, M.,
Pecorelli, S., Peracino, A.P., Quaranta, N., Stella, R., Lin, F.R., 2014. Hearing loss and
cognitive decline in older adults: questions and answers. Aging Clin. Exp. Res. 26,
567–573. https://doi.org/10.1007/s40520-014-0266-3.
Bertenthal, B.I., Fischer, K.W., 1978. Development of self-recognition in the infant. Dev.
Psychol. 14, 44–50. https://doi.org/10.1037/0012-1649.14.1.44.
Blanchower, D.G., Oswald, A.J., 2008. Is well-being U-shaped over the life cycle? Soc.
Sci. Med. 66, 1733–1749. https://doi.org/10.1016/j.socscimed.2008.01.030.
Bogin, B., 1997. Evolutionary hypotheses for human childhood. Am. J. Phys. Anthropol.
104, 63–89. https://doi.org/10.1002/(sici)1096-8644(1997)25+<63::aid-
ajpa3>3.0.co;2-8.
Brinks, R., Landwehr, S., Waldeyer, R., 2013. Age of onset in chronic diseases: new
method and application to dementia in Germany. Popul. Health Metrics 11. https://
doi.org/10.1186/1478-7954-11-6.
Centers for Disease Control and Prevention, 2017. Mortality in the United States, 2016.
https://www.cdc.gov/nchs/data/databriefs/db293.pdf.
Chronaki, G., Hadwin, J.A., Garner, M., Maurage, P., Sonuga-Barke, E.J., 2014. The
development of emotion recognition from facial expressions and non-linguistic vo-
calizations during childhood. Br. J. Dev. Psychol. 33, 218–236. https://doi.org/
10.1111/bjdp.12075.
De Bruin, J.P., te Velde, E.R., 2004. Female reproductive ageing: concepts and conse-
quences. In: Tulandi, T., Gosden, R.G. (Eds.), Preservation of Fertility. Taylor &
Francis, London, pp. 1–20.
De S´
a, C.D., Bofno, C.C., Ramos, R.T., Tanaka, C., 2018. Development of postural
control and maturation of sensory systems in children of different ages a cross-
sectional study. Braz. J. Phys. Ther. 22, 70–76. https://doi.org/10.1016/j.
bjpt.2017.10.006.
Del Giudice, M., 2009. Sex, attachment, and the development of reproductive strategies.
Behav. Brain Sci. 32, 1–21. https://doi.org/10.1017/s0140525x09000016.
Demetrius, L., Legendre, S., Harrem¨
oes, P., 2009. Evolutionary entropy: a predictor of
body size, metabolic rate and maximal life span. Bull. Math. Biol. 71, 800–818.
https://doi.org/10.1007/s11538-008-9382-6.
Dutton, E., Madison, G., van der Linden, D., 2021. Genius and premature birth: little
evidence that claims about historically eminent scientists are accurate. Indian J.
Hist. Sci. 56, 20–27. https://doi.org/10.1007/s43539-021-00005-1.
Eddington, A.S., 1929. The Nature of the Physical World. Univ. Press, Cambridge.
Edelstein, R.S., Chopik, W.J., Kean, E.L., 2011. Sociosexuality moderates the association
between testosterone and relationship status in men and women. Horm. Behav. 60,
248–255. https://doi.org/10.1016/j.yhbeh.2011.05.007.
Ehrenberg, M., Elf, J., Aurell, E., Sandberg, R., Tegn´
er, J., 2003. Systems biology is taking
off. Genome Res. 13, 2377–2380. https://doi.org/10.1101/gr.1763203.
Eijkemans, M.J.C., van Poppel, F., Habbema, D.F., Smith, K.R., Leridon, H., te Velde, E.
R., 2014. Too old to have children? lessons from natural fertility populations. Hum.
Reprod. 29, 1304–1312. https://doi.org/10.1093/humrep/deu056.
Ellemberg, D., Lewis, T.L., Hong Liu, C., Maurer, D., 1999. Development of spatial and
temporal vision during childhood. Vis. Res. 39, 2325–2333. https://doi.org/
10.1016/s0042-6989(98)00280-6.
Falangola, M.F., Jensen, J.H., Babb, J.S., Hu, C., Castellanos, F.X., Di Martino, A.,
Ferris, S.H., Helpern, J.A., 2008. Age-related non-Gaussian Diffusion Patterns in the
prefrontal brain. J. Magn. Reson. Imag. 28, 1345–1350. https://doi.org/10.1002/
jmri.21604.
Feinberg, I., Campbell, I.G., 2010. The onset of the adolescent delta power decline occurs
after age 11 years: a comment on Tarokh and Carskadon. Sleep 33, 737. https://doi.
org/10.1093/sleep/33.6.737, 737.
Ferring, V., Pancherz, H., 2008. Divine proportions in the growing face. Am. J. Orthod.
Dentofacial Orthop. 134, 472–479. https://doi.org/10.1016/j.ajodo.2007.03.027.
Finer, L.B., Philbin, J.M., 2014. Trends in ages at key reproductive transitions in the
United States. Wom. Health Issues 24, 1951–2010. https://doi.org/10.1016/j.
whi.2014.02.002.
Fraiman, D., Chialvo, D.R., 2012. What kind of noise is brain noise: anomalous scaling
behavior of the resting brain activity uctuations. Front. Physiol. 3 https://doi.org/
10.3389/fphys.2012.00307.
Freeman, E.W., 2010. Associations of depression with the transition to Menopause.
Menopause 17, 823–827. https://doi.org/10.1097/gme.0b013e3181db9f8b.
Gallup, G.G., 1970. Chimpanzees: self-recognition. Science 167, 86–87. https://doi.org/
10.1126/science.167.3914.86.
Germine, L.T., Duchaine, B., Nakayama, K., 2011. Where cognitive development and
aging meet: face learning ability peaks after age 30. Cogn 118, 201–210. https://doi.
org/10.1016/j.cognition.2010.11.002.
Giedd, J.N., 2004. Structural magnetic resonance imaging of the adolescent brain. Ann.
N. Y. Acad. Sci. 1021, 77–85. https://doi.org/10.1196/annals.1308.009.
Giedd, J.N., Blumenthal, J., Jeffries, N.O., Castellanos, F.X., Liu, H., Zijdenbos, A.,
Paus, T., Evans, A.C., Rapoport, J.L., 1999. Brain development during childhood and
adolescence: a longitudinal MRI study. Nat. Neurosci. 2, 861–863. https://doi.org/
10.1038/13158.
Goldberger, A.L., West, B.J., Dresselhaus, T., Bhargava, V., 1985. Bronchial asymmetry
and Fibonacci scaling. Exp 41, 1537–1538. https://doi.org/10.1007/bf01964794.
Hafkemeijer, A., Altmann-Schneider, I., Craen, A.J., Slagboom, P.E., Grond, J.,
Rombouts, S.A., 2014. Associations between age and gray matter volume in
Anatomical Brain Networks in middle-aged to older adults. Aging Cell 13,
1068–1074. https://doi.org/10.1111/acel.12271.
Hartshorne, J.K., Germine, L.T., 2015. When does cognitive functioning peak? the
asynchronous rise and fall of different cognitive abilities across the life span. Psy-
chol. Sci. 26, 433–443. https://doi.org/10.1177/0956797614567339.
Hawking, S., 1988. A Brief History of Time. Bantam Books, New York.
Hiddinga, A.E., Van Den Hoofdakker, R.H., 1997. Endogenous and exogenous compo-
nents in the circadian variation of core body temperature in humans. J. Sleep Res. 6,
156–163. https://doi.org/10.1046/j.1365-2869.1997.00047.x.
Hochberg, Z., 2008. Juvenility in the context of life history theory. Arch. Dis. Child. 93,
534–539. https://doi.org/10.1136/adc.2008.137570.
Hodges, J., Cao, S., Cladis, D., Weaver, C., 2019. Lactose intolerance and Bone Health:
the challenge of ensuring adequate calcium intake. Nutrients 11, 718. https://doi.
org/10.3390/nu11040718.
Iosa, M., Morone, G., Paolucci, S., 2017. Golden Gait: an optimization theory perspective
on human and humanoid walking. Front. Neurorob. 11 https://doi.org/10.3389/
fnbot.2017.00069.
Jean, R.V., 1994. Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge
Univ. Press, Cambridge.
Kail, R.V., 2022. Human Development: A Life-Span View. Cengage Learning.
Kelsey, T.W., Li, L.Q., Mitchell, R.T., Whelan, A., Anderson, R.A., Wallace, W.H., 2014.
A validated age-related normative model for male total testosterone shows
increasing variance but no decline after age 40 years. PLoS One 9. https://doi.org/
10.1371/journal.pone.0109346.
Kirkwood, T.B., Cremer, T., 1982. Cytogerontology since 1881: a reappraisal of august
weismann and a review of modern progress. Hum. Genet. 60, 101–121. https://doi.
org/10.1007/bf00569695.
Kitano, H., 2002. Systems biology: a brief overview. Science 295, 1662–1664. https://
doi.org/10.1126/science.1069492.
Klieger, D.M., 2013. Saunders Essentials of Medical Assisting - E-Book. Saunders.
Knickmeyer, R.C., Gouttard, S., Kang, C., Evans, D., Wilber, K., Smith, J.K., Hamer, R.M.,
Lin, W., Gerig, G., Gilmore, J.H., 2008. A structural MRI study of human brain
development from birth to 2 years. J. Neurosci. 28, 12176–12182. https://doi.org/
10.1523/jneurosci.3479-08.2008.
Lagarkova, M.A., Eremeev, A.V., Svetlakov, A.V., Rubtsov, N.B., Kiselev, S.L., 2010.
Human embryonic stem cell lines isolation, cultivation, and characterization. In
Vitro Cell. Dev. Biol. Anim. 46, 284–293. https://doi.org/10.1007/s11626-010-
9282-6.
Larsson, L., Grimby, G., Karlsson, J., 1979. Muscle Strength and speed of movement in
relation to age and muscle morphology. J. Appl. Physiol. 46, 451–456. https://doi.
org/10.1152/jappl.1979.46.3.451.
Lewis, M., Brooks-Gunn, J., 1979. Toward a theory of social cognition: the development
of self. Child Adolesc. Dev. 1–20. https://doi.org/10.1002/cd.23219790403. New
Dir.
Linage, G., Montoya, F., Sarmiento, A., Showalter, K., Parmananda, P., 2006. Fibonacci
order in the period-doubling cascade to chaos. Phys. Lett. 359, 638–639. https://doi.
org/10.1016/j.physleta.2006.07.036.
Livio, M., 2008. The Golden Ratio: the Story of Phi, the World’s Most Astonishing
Number. Broadway Books, New York.
Loring, D.W., Lee, G.P., Meador, K.J., 1989. Issues in memory assessment of the elderly.
Clin. Geriatr. Med. 5, 565–581. https://doi.org/10.1016/s0749-0690(18)30671-2.
Luebberding, S., Krueger, N., Kerscher, M., 2014. Quantication of age-related facial
wrinkles in men and women using a three-dimensional fringe projection method and
validated assessment scales. Dermatol. Surg. 40, 22–32. https://doi.org/10.1111/
dsu.12377.
Marcer, P., Rowlands, P., 2017. Nilpotent quantum mechanics: analogs and applications.
Front. Physiol. 5 https://doi.org/10.3389/fphy.2017.00028.
Mathews, T.J., Mini˜
no, A.M., Osterman, M.J., Strobino, D.M., Guyer, B., 2011. Annual
summary of vital statistics: 2008. An. Pediatr. 127, 146–157. https://doi.org/
10.1542/peds.2010-3175.
Minors, D.S., Waterhouse, J.M., 1981. Circadian Rhythms and the Human. Wright. PSG,
Bristol etc.
Nelson, K., 1992. Emergence of autobiographical memory at age 4. Hum. Dev. 35,
172–177. https://doi.org/10.1159/000277149.
Nelson, S.J., 2020. Wheeler’s Dental Anatomy, Physiology, and Occlusion. Elsevier, St.
Louis, MO.
Nelson, W.L., 1983. Physical principles for economies of skilled movements. Biol.
Cybern. 46, 135–147. https://doi.org/10.1007/bf00339982.
Nielsen, M., Dissanayake, C., 2004. Pretend play, mirror self-recognition and imitation: a
longitudinal investigation through the second year. Infant Behav. Dev. 27, 342–365.
https://doi.org/10.1016/j.infbeh.2003.12.006.
Paul, S.P., 2016. Golden spirals and scalp whorls: nature’s own design for rapid
expansion. PLoS One 11. https://doi.org/10.1371/journal.pone.0162026.
Perez, J.-C., 2010. Codon populations in single-stranded whole human genome DNA are
fractal and ne-tuned by the golden ratio 1.618. Interdiscip. Sci. 2, 228–240.
https://doi.org/10.1007/s12539-010-0022-0.
Piaget, J., 1952. The Origins of Intelligence in Children. International Universities Press,
New York.
Prigogine, I., Stengers, I., 1984. Order Out of Chaos. Batam Books, Toronto.
Roberts, G.J., Lucas, V.S., Andiappan, M., McDonald, F., 2016. Dental age estimation:
pattern recognition of root canal widths of mandibular molars. A novel mandibular
maturity marker at the 18-year threshold. J. Forensic Sci. 62, 351–354. https://doi.
org/10.1111/1556-4029.13287.
Robinson, E., Almiron-Roig, E., Rutters, F., de Graaf, C., Forde, C.G., Tudur Smith, C.,
Nolan, S.J., Jebb, S.A., 2014. A systematic review and meta-analysis examining the
effect of eating rate on energy intake and hunger. Am. J. Clin. Nutr. 100, 123–151.
https://doi.org/10.3945/ajcn.113.081745.
Roopun, A.K., 2008. Temporal interactions between cortical rhythms. Front. Neurosci. 2,
145–154. https://doi.org/10.3389/neuro.01.034.2008.
Rosselli, M., Ardila, A., Matute, E., V´
elez-Uribe, I., 2014. Language development across
the life span: a neuropsychological/neuroimaging perspective. Neurosci. J. 1–21.
https://doi.org/10.1155/2014/585237, 2014.
R.G. Sacco and J.S. Torday
Progress in Biophysics and Molecular Biology 177 (2023) 24–33
33
Rossi, A., 2004. In: Brim, O.G., Kessler, R.C., Ryff, C.D. (Eds.), How Healthy Are We?: A
National Study of Well-Being at Midlife. University of Chicago Press.
Rubin, D.C., 2000. The distribution of early childhood memories. MEM (Miner. Elec-
trolyte Metab.) 8, 265–269. https://doi.org/10.1080/096582100406810.
Sacco, R.G., 2013. Re-envisaging the eight developmental stages of Erik Erikson: the
Fibonacci life-chart method (FLCM). J. Educ. Develop. Psychol. 3 https://doi.org/
10.5539/jedp.v3n1p140.
Sacco, R.G., 2017. The Fibonacci life-chart method as a predictor of spiritual experience.
J. Educ. Develop. Psychol. 7, 1. https://doi.org/10.5539/jedp.v7n2p1.
Sacco, R.G., 2019a. Modeling celestial mechanics using the Fibonacci numbers. Int. J.
Astron. 8, 8–12. https://doi.org/10.5923/j.astronomy.20190801.02.
Sacco, R.G., 2019b. The predictability of synchronicity experience: results from a survey
of Jungian analysts. Int. J. Psychol. Stud. 11, 46. https://doi.org/10.5539/ijps.
v11n3p46.
Sagan, L., 1967. On the origin of mitosing cells. J. Theor. Biol. 14 https://doi.org/
10.1016/0022-5193(67)90079-3.
Schoenaker, D.A.J.M., Jackson, C.A., Rowlands, J.V., Mishra, G.D., 2014. Socioeconomic
position, lifestyle factors and age at Natural Menopause: a systematic review and
meta-analyses of studies across six continents. Int. J. Epidemiol. 43, 1542–1562.
https://doi.org/10.1093/ije/dyu094.
Schr¨
odinger, E., 1967. What Is Life? the Physical Aspects of the Living Cell: Mind and
Matter. Cambridge University Press, Cambridge, UK.
Sharp, W.E., 1971. An analysis of the laws of stream order for Fibonacci drainage pat-
terns. Water Res. Educ. 7, 1548–1557. https://doi.org/10.1029/wr007i006p01548.
Silva, C., Annamalai, K., 2008. Entropy generation and human aging: lifespan entropy
and effect of physical activity level. Entropy 10, 100–123. https://doi.org/10.3390/
entropy-e10020100.
Silva, C.A., Annamalai, K., 2009. Entropy generation and human aging: lifespan entropy
and effect of diet composition and caloric restriction diets. J. Thermodyn. 1–10.
https://doi.org/10.1155/2009/186723, 2009.
Sokoloff, L., 1999. Energetics of functional activation in neural tissues. Neurochem. Res.
24, 321–329. https://doi.org/10.1023/a:1022534709672.
Stagi, S., Cavalli, L., Iurato, C., Seminara, S., Brandi, M.L., de Martino, M., 2013. Bone
metabolism in children and adolescents: main characteristics of the determinants of
peak bone mass. Clin. Cases. Miner. Bone Metab. 10 (3), 172–179.
Stam, C.J., de Bruin, E.A., 2004. Scale-free dynamics of global functional connectivity in
the human brain. Hum. Brain Mapp. 22, 97–109. https://doi.org/10.1002/
hbm.20016.
Stapel, J.C., van Wijk, I., Bekkering, H., Hunnius, S., 2016. Eighteen-month-old infants
show distinct electrophysiological responses to their own faces. Decis. Sci. J. 20
https://doi.org/10.1111/desc.12437.
Stolk, L., Perry, J.R., Chasman, D.I., He, C., Mangino, M., Sulem, P., et al., 2012. Meta-
analyses identify 13 loci associated with age at menopause and highlight DNA repair
and immune pathways. Nat. Genet. 44, 260–268. https://doi.org/10.1038/ng.1051.
Størvold, G.V., Aarethun, K., Bratberg, G.H., 2013. Age for onset of walking and pre-
walking strategies. Early Hum. Dev. 89, 655–659. https://doi.org/10.1016/j.
earlhumdev.2013.04.010.
Terracciano, A., Costa, P.T., McCrae, R.R., 2006. Personality plasticity after age 30. Pers.
Soc. Psychol. Bull. 32, 999–1009. https://doi.org/10.1177/0146167206288599.
Tomasi, D., Wang, G.-J., Volkow, N.D., 2013. Energetic cost of brain functional con-
nectivity. Proc. Natl. Acad. Sci. USA 110, 13642–13647. https://doi.org/10.1073/
pnas.1303346110.
Torday, J., 2016. The cell as the rst niche construction. Biol. 5, 19. https://doi.org/
10.3390/biology5020019.
Torday, J.S., 2015. A central theory of Biology. Med. Hypotheses 85, 49–57. https://doi.
org/10.1016/j.mehy.2015.03.019.
Torday, J.S., 2018. Quantum mechanics predicts evolutionary biology. Prog. Biophys.
Mol. Biol. 135, 11–15. https://doi.org/10.1016/j.pbiomolbio.2018.01.003.
Torday, J.S., 2019a. Cell-cell communication predicts aging, senescence and death: an
integrated, predictive evolutionary approach. Biomed. Rev. 30, 15. https://doi.org/
10.14748/bmr.v30.6383.
Torday, J.S., 2019b. The singularity of nature. Prog. Biophys. Mol. Biol. 142, 23–31.
https://doi.org/10.1016/j.pbiomolbio.2018.07.013.
Torday, J.S., 2020. Consciousness, redux. Med. Hypotheses 140, 109674. https://doi.
org/10.1016/j.mehy.2020.109674.
Torday, J.S., 2021. Life is a mobius strip. Prog. Biophys. Mol. Biol. 167, 41–45. https://
doi.org/10.1016/j.pbiomolbio.2021.08.001.
Torday, J.S., Miller, W.B., 2018. The cosmologic continuum from physics to conscious-
ness. Prog. Biophys. Mol. Biol. 140, 41–48. https://doi.org/10.1016/j.
pbiomolbio.2018.04.005.
Torday, J.S., Rehan, V.K., 2009. Lung evolution as a cipher for physiology. Physiol.
Genom. 38, 1–6. https://doi.org/10.1152/physiolgenomics.90411.2008.
Torday, J.S., Rehan, V.K., 2011. A cell-molecular approach predicts vertebrate evolution.
Mol. Biol. Evol. 28, 2973–2981. https://doi.org/10.1093/molbev/msr134.
Vary, J.C., 2015. Selected disorders of skin appendages—acne, alopecia, hyperhidrosis.
Med. Clin. 99, 1195–1211. https://doi.org/10.1016/j.mcna.2015.07.003.
Verguts, J., Ameye, L., Bourne, T., Timmerman, D., 2013. Normative data for uterine size
according to age and gravidity and possible role of the classical golden ratio. Ul-
trasound Obstet. Gynecol. 42, 713–717. https://doi.org/10.1002/uog.12538.
Volbert, R., 2000. Sexual knowledge of preschool children. J. Psychol. Hum. Sex. 12,
5–26. https://doi.org/10.1300/j056v12n01_02.
Wehrl, A., 1978. General properties of entropy. Rev. Mod. Phys. 50, 221–260. https://
doi.org/10.1103/revmodphys.50.221.
Weiss, H., Weiss, V., 2003. The golden mean as clock cycle of brain waves. Chaos, Solit.
Fractals 18, 643–652. https://doi.org/10.1016/s0960-0779(03)00026-2.
Weon, B.M., Je, J.H., 2008. Theoretical estimation of maximum human lifespan. Bio-
gerontology 10, 65–71. https://doi.org/10.1007/s10522-008-9156-4.
Werker, J.F., Hensch, T.K., 2015. Critical periods in speech perception: new Directions.
Annu. Rev. Psychol. 66, 173–196. https://doi.org/10.1146/annurev-psych-010814-
015104.
Werner, G., 2010. Fractals in the nervous system: conceptual implications for theoretical
neuroscience. Front. Physiol. https://doi.org/10.3389/fphys.2010.00015.
Werner, L., 2020. Human Auditory Development. Routledge.
West, B.J., 1990. Physiology in fractal dimensions: error tolerance. Ann. Biomed. Eng.
18, 135–149. https://doi.org/10.1007/bf02368426.
West, G.B., Brown, J.H., 2005. The origin of allometric scaling laws in biology from
genomes to ecosystems: towards a quantitative unifying theory of biological struc-
ture and organization. J. Exp. Biol. 208, 1575–1592. https://doi.org/10.1242/
jeb.01589.
West, G.B., Woodruff, W.H., Brown, J.H., 2002. Allometric scaling of metabolic rate from
molecules and mitochondria to cells and mammals. Proc. Natl. Acad. Sci. USA 99,
2473–2478. https://doi.org/10.1073/pnas.012579799.
Weston, A.D., Hood, L., 2004. Systems Biology, proteomics, and the future of Health
Care: toward predictive, preventative, and personalized medicine. J. Proteome Res.
3, 179–196. https://doi.org/10.1021/pr0499693.
Yamagishi, M.E., Shimabukuro, A.I., 2007. Nucleotide frequencies in human genome and
Fibonacci numbers. Bull. Math. Biol. 70, 643–653. https://doi.org/10.1007/s11538-
007-9261-6.
Yetkin, E., Topbas
¸, U., Yanik, A., Yetkin, G., 2014. Does systolic and diastolic blood
pressure follow golden ratio? Int. J. Cardiol. 176, 1457–1459. https://doi.org/
10.1016/j.ijcard.2014.08.065.
Robert G. Sacco
Independent Researcher, Toronto, Canada
E-mail address: robgsacco@gmail.com.
John S. Torday
*
Pediatrics, Obstetrics and Gynecology, Evolutionary Medicine, University of
California, Los Angeles, Westwood Boulevard, Westwood, CA, 90095, USA
*
Corresponding author. Obstetrics and Gynecology, Evolutionary
Medicine, University of California, Los Angeles, Westwood Boulevard,
Westwood, CA, 90095, USA.
E-mail address: jtorday@ucla.edu (J.S. Torday).
R.G. Sacco and J.S. Torday
