Among other examples, the theorisation of ectopic action potentials in neurons using a Kramers-Moyal expansion and the description of physical phenomena measured during an EEG using a dipole approximation use neurophysics to better understand neural activity.
Another quite distinct theoretical approach considers neurons as having Ising model energies of interaction and explores the physical consequences of this for various Cayley tree topologies and large neural networks. In 1981, the exact solution for the closed Cayley tree (with loops) was derived by Peter Barth for an arbitrary branching ratio and found to exhibit an unusual phase transition behavior in its local-apex and long-range site-site correlations, suggesting that the emergence
of structurally-determined and connectivity-influenced cooperative
phenomena may play a significant role in large neural networks.
Recording techniques
Old techniques to record brain activity using physical phenomena are already widespread in research and medicine. Electroencephalography (EEG) uses electrophysiology to measure electrical activity within the brain. This technique, with which Hans Berger first recorded brain electrical activity on a human in 1924, is non-invasive and uses electrodes placed on the scalp of the patient to record brain activity. Based on the same principle, electrocorticography (ECoG) requires a craniotomy to record electrical activity directly on the cerebral cortex.
In the recent decades, physicists have come up with technologies and devices to image the brain and its activity. The Functional Magnetic Resonance Imaging (fMRI) technique, discovered by Seiji Ogawa in 1990, reveals blood flow changes inside the brain. Based on the existing medical imaging technique Magnetic Resonance Imaging
(MRI) and on the link between the neural activity and the cerebral
blood flow, this tool enables scientists to study brain activities when
they are triggered by a controlled stimulation. Another technique, the Two Photons Microscopy (2P), invented by Winfried Denk (for which he has been awarded the Brain Prize in 2015), John H. Strickler and Watt W. Webb in 1990 at Cornell University, uses fluorescent proteins and dyes to image brain cells. This technique combines the two-photon absorption, first theorized by Maria Goeppert-Mayer in 1931, with lasers. Today, this technique is widely used in research and often coupled with genetic engineering to study the behavior of a specific type of neuron.
Theories of consciousness
Consciousness
is still an unknown mechanism and theorists have yet to come up with
physical hypotheses explaining its mechanisms. Some theories rely on the
idea that consciousness could be explained by the disturbances in the
cerebral electromagnetic field generated by the action potentials triggered during brain activity. These theories are called electromagnetic theories of consciousness. Another group of hypotheses suggest that consciousness cannot be explained by classical dynamics but with quantum mechanics and its phenomena. These hypotheses are grouped into the idea of quantum mind and were first introduced by Eugene Wigner.
Among
the list of prizes that reward neurophysicists for their contribution
to neurology and related fields, the most notable one is the Brain Prize, whose last laureates are Adrian Bird and Huda Zoghbi
for "their groundbreaking work to map and understand epigenetic
regulation of the brain and for identifying the gene that causes Rett
syndrome". The other most relevant prizes that can be awarded to a neurophysicist are: the NAS Award in the Neurosciences, the Kavli Prize and to some extent the Nobel Prize in Physiology or Medicine.
It can be noted that a Nobel Prize was awarded to scientists that
developed techniques which contributed widely to a better understanding
of the nervous system, such as Neher and Sakmann in 1991 for the patch clamp, and also to Lauterbur and Mansfield for their work on Magnetic resonance imaging (MRI) in 2003.
Molecular biophysics typically addresses biological questions similar to those in biochemistry and molecular biology,
seeking to find the physical underpinnings of biomolecular phenomena.
Scientists in this field conduct research concerned with understanding
the interactions between the various systems of a cell, including the
interactions between DNA, RNA and protein biosynthesis, as well as how these interactions are regulated. A great variety of techniques are used to answer these questions.
In addition to traditional (i.e. molecular and cellular) biophysical topics like structural biology or enzyme kinetics, modern biophysics encompasses an extraordinarily broad range of research, from bioelectronics to quantum biology
involving both experimental and theoretical tools. It is becoming
increasingly common for biophysicists to apply the models and
experimental techniques derived from physics, as well as mathematics and statistics, to larger systems such as tissues, organs, populations and ecosystems. Biophysical models are used extensively in the study of electrical conduction in single neurons, as well as neural circuit analysis in both tissue and whole brain.
William T. Bovie (1882–1958) is credited as a leader of the field's further development in the mid-20th century. He was a leader in developing electrosurgery.
The popularity of the field rose when the book What Is Life? by Erwin Schrödinger was published. Since 1957, biophysicists have organized themselves into the Biophysical Society which now has about 9,000 members over the world.
Some authors such as Robert Rosen
criticize biophysics on the ground that the biophysical method does not
take into account the specificity of biological phenomena.
Focus as a subfield
While
some colleges and universities have dedicated departments of
biophysics, usually at the graduate level, many do not have
university-level biophysics departments, instead having groups in
related departments such as biochemistry, cell biology, chemistry, computer science, engineering, mathematics, medicine, molecular biology, neuroscience, pharmacology, physics, and physiology.
Depending on the strengths of a department at a university differing
emphasis will be given to fields of biophysics. What follows is a list
of examples of how each department applies its efforts toward the study
of biophysics. This list is hardly all inclusive. Nor does each subject
of study belong exclusively to any particular department. Each academic
institution makes its own rules and there is much overlap between
departments.
Medicine
– biophysical research that emphasizes medicine. Medical biophysics is
a field closely related to physiology. It explains various aspects and
systems of the body from a physical and mathematical perspective.
Examples are fluid dynamics
of blood flow, gas physics of respiration, radiation in
diagnostics/treatment and much more. Biophysics is taught as a
preclinical subject in many medical schools, mainly in Europe.
Neuroscience – studying neural networks experimentally (brain slicing) as well as theoretically (computer models), membrane permittivity.
Quantum biology – The field of quantum biology applies quantum mechanics to biological objects and problems. Decoheredisomers to yield time-dependent base substitutions. These studies imply applications in quantum computing.
Many biophysical techniques
are unique to this field. Research efforts in biophysics are often
initiated by scientists who were biologists, chemists or physicists by
training.
Evolutionary ecology lies at the intersection of ecology and evolutionary biology.
It approaches the study of ecology in a way that explicitly considers
the evolutionary histories of species and the interactions between them.
Conversely, it can be seen as an approach to the study of evolution
that incorporates an understanding of the interactions between the
species under consideration. The main subfields of evolutionary ecology
are life history evolution, sociobiology (the evolution of social behavior), the evolution of interspecific interactions (e.g. cooperation, predator–prey interactions, parasitism, mutualism) and the evolution of biodiversity and of ecological communities.
Evolutionary ecology mostly considers two things: how
interactions (both among species and between species and their physical
environment) shape species through selection and adaptation, and the
consequences of the resulting evolutionary change.
Evolutionary models
A large part of evolutionary ecology is about utilising models and finding empirical data as proof. Examples include the Lack clutch size model devised by David Lack and his study of Darwin's finches
on the Galapagos Islands. Lack's study of Darwin's finches was
important in analyzing the role of different ecological factors in speciation. Lack suggested that differences in species were adaptive and produced by natural selection, based on the assertion by G.F. Gause that two species cannot occupy the same niche.
Richard Levins
introduced his model of the specialization of species in 1968, which
investigated how habitat specialization evolved within heterogeneous
environments using the fitness sets an organism or species possesses.
This model developed the concept of spatial scales in specific
environments, defining fine-grained spatial scales and coarse-grained
spatial scales.
The implications of this model include a rapid increase in
environmental ecologists' understanding of how spatial scales impact
species diversity in a certain environment.
Another model is Law and Diekmann's 1996 models on mutualism, which is defined as a relationship between two organisms that benefits both individuals.
Law and Diekmann developed a framework called adaptive dynamics, which
assumes that changes in plant or animal populations in response to a
disturbance or lack thereof occurs at a faster rate than mutations
occur. It is aimed to simplify other models addressing the relationships
within communities.
Tangled nature model
The
tangled nature model provides different methods for demonstrating and
predicting trends in evolutionary ecology. The model analyzes an
individual prone to mutation within a population as well as other factors such as extinction rate.
The model was developed by Simon Laird, Daniel Lawson, and Henrik
Jeldtoft Jensen of the Imperial College London in 2002. The purpose of
the model is to create a simple and logical ecological model based on
observation. The model is designed such that ecological effects can be
accounted for when determining form, and fitness of a population.
Ecological genetics tie into evolutionary ecology through the study of how traits evolve in natural populations.
Ecologists are concerned with how the environment and timeframe leads
to genes becoming dominant. Organisms must continually adapt in order to
survive in natural habitats. Genes define which organisms survive and
which will die out. When organisms develop different genetic variations,
even though they stem from the same species, it is known as
polymorphism. Organisms that pass on beneficial genes continue to evolve their species to have an advantage inside of their niche.
The basis of the central principles of evolutionary ecology can be attributed to Charles Darwin (1809–1882), specifically in referencing his theory of natural selection and population dynamics, which discusses how populations of a species change over time. According to Ernst Mayr,
professor of zoology at Harvard University, Darwin's most distinct
contributions to evolutionary biology and ecology are as follows: "The
first is the non-constancy of species, or the modern conception of evolution
itself. The second is the notion of branching evolution, implying the
common descent of all species of living things on earth from a single
unique origin."
Additionally, "Darwin further noted that evolution must be gradual,
with no major breaks or discontinuities. Finally, he reasoned that the
mechanism of evolution was natural selection."
George Evelyn Hutchinson
George Evelyn Hutchinson's
(1903–1991) contributions to the field of ecology spanned over 60
years, in which he had significant influence in systems ecology,
radiation ecology, limnology, and entomology. Described as the "father of modern ecology" by Stephen Jay Gould,
Hutchinson was one of the first scientists to link the subjects of
ecology and mathematics. According to Hutchinson, he constructed
"mathematical models of populations, the changing proportions of
individuals of various ages, birthrate, the ecological niche, and population interaction in this technical introduction to population ecology." He also had a vast interest in limnology, due to his belief that lakes could be studied as a microcosm that provides insight into system behavior.
Hutchinson is also known for his work Circular Causal Systems in
Ecology, in which he states that "groups of organisms may be acted upon
by their environment, and they may react upon it. If a set of properties
in either system changes in such a way that the action of the first
system on the second changes, this may cause changes in properties of
the second system which alter the mode of action of the second system on
the first."
Robert MacArthur
Robert MacArthur (1930–1972) is best known in the field of Evolutionary Ecology for his work The Theory of Island Biogeography,
in which he and his co-author propose "that the number of species on
any island reflects a balance between the rate at which new species
colonize it and the rate at which populations of established species
become extinct."
Eric Pianka
According to the University of Texas, Eric Pianka's
(1939–2022) work in evolutionary ecology includes foraging strategies,
reproductive tactics, competition and niche theory, community structure
and organization, species diversity, and understanding rarity.
Pianka is also known for his interest in lizards to study ecological
occurrences, as he claimed they were "often abundant, making them
relatively easy to locate, observe, and capture."
Michael Rosenzweig
Michael L. Rosenzweig (1941–present) created and popularized Reconciliation ecology, which began with his theory that designated nature preserves would not be enough land to conserve the biodiversity of Earth, as humans have used so much land that they have negatively impacted biogeochemical cycles and had other ecological impacts that have negatively affected species compositions.
Thierry Lodé (1956–present), a French ecologist whose work focused on how sexual conflict in populations of species impacts evolution.
Research
Michael Rosenzweig's idea of reconciliation ecology was developed based on existing research, which was conducted on the principle first suggested by Alexander von Humboldt stating that larger areas of land will have increased species diversity as compared to smaller areas. This research focused on species-area relationships
(SPARs) and the different scales on which they exist, ranging from
sample-area to interprovincial SPARs. Steady-state dynamics in diversity
gave rise to these SPARs, which are now used to measure the reduction
of species diversity on Earth. In response to this decline in diversity,
Rosenzweig's reconciliation ecology was born.
Evolutionary ecology has been studied using symbiotic
relationships between organisms to determine the evolutionary forces by
which such relationships develop. In symbiotic relationships, the symbiont
must confer some advantage to its host in order to persist and continue
to be evolutionarily viable. Research has been conducted using aphids
and the symbiotic bacteria with which they coevolve. These bacteria are
most frequently conserved from generation to generation, displaying high
levels of vertical transmission.
Results have shown that these symbiotic bacteria ultimately confer some
resistance to parasites to their host aphids, which both increases the
fitness of the aphids and lead to symbiont-mediated coevolution between
the species.
Color variation in cichlid fish
The effects of evolutionary ecology and its consequences can be seen in the case of color variation among African cichlid fish. With over 2,000 species, cichlid fishes are very species-rich and capable of complex social interactions. Polychromatism,
the variation of color patterns within a population, occurs within
cichlid fishes due to environmental adaptations and to increase chances
of sexual reproduction.
From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Quantum_evolution Quantum evolution is a component of George Gaylord Simpson's multi-tempoed theory of evolution proposed to explain the rapid emergence of higher taxonomic groups in the fossil record. According to Simpson, evolutionary rates
differ from group to group and even among closely related lineages.
These different rates of evolutionary change were designated by Simpson
as bradytelic (slow tempo), horotelic (medium tempo), and tachytelic (rapid tempo).
Quantum evolution differed from these styles of change in that it involved a drastic shift in the adaptive zones of certain classes of animals. The word "quantum"
therefore refers to an "all-or-none reaction", where transitional forms
are particularly unstable, and thereby perish rapidly and completely. Although quantum evolution may happen at any taxonomic level, it plays a much larger role in "the origin taxonomic units of relatively high rank, such as families, orders, and classes."
Quantum evolution in plants
Usage of the phrase "quantum evolution" in plants was apparently first articulated by Verne Grant in 1963 (pp. 458-459). He cited an earlier 1958 paper by Harlan Lewis and Peter H. Raven,
wherein Grant asserted that Lewis and Raven gave a "parallel"
definition of quantum evolution as defined by Simpson. Lewis and Raven
postulated that species in the Genus Clarkia had a mode of speciation that resulted
...as
a consequence of a rapid reorganization of the chromosomes due to the
presence, at some time, of a genotype conducive to extensive chromosome
breakage. A similar mode of origin by rapid reorganization of the
chromosomes is suggested for the derivation of other species of Clarkia.
In all of these examples the derivative populations grow adjacent to
the parental species, which they resemble closely in morphology, but
from which they are reproductively isolated because of multiple
structural differences in their chromosomes. The spatial relationship of
each parental species and its derivative suggests that differentiation
has been recent. The repeated occurrence of the same pattern of
differentiation in Clarkia suggests that a rapid reorganization of
chromosomes has been an important mode of evolution in the genus. This
rapid reorganization of the chromosomes is comparable to the systemic
mutations proposed by Goldschmidt as a mechanism of macroevolution. In Clarkia,
we have not observed marked changes in physiology and pattern of
development that could be described as macroevolution. Reorganization of
the genomes may, however, set the stage for subsequent evolution along a
very different course from that of the ancestral populations
Harlan Lewis refined this concept in a 1962 paper
where he coined the term "Catastrophic Speciation" to describe this
mode of speciation, since he theorized that the reductions in population
size and consequent inbreeding that led to chromosomal rearrangements
occurred in small populations that were subject to severe drought.
we
can define quantum speciation as the budding off of a new and very
different daughter species from a semi-isolated peripheral population of
the ancestral species in a cross-fertilizing organism...as compared
with geographical speciation, which is a gradual and conservative
process, quantum speciation is rapid and radical in its phenotypic or
genotypic effects or both.
Gottlieb did not believe that sympatric speciation required disruptive selection
to form a reproductive isolating barrier, as defined by Grant, and in
fact Gottlieb stated that requiring disruptive selection was
"unnecessarily restrictive"
in identifying cases of sympatric speciation. In this 2003 paper
Gottlieb summarized instances of quantum evolution in the plant species Clarkia, Layia, and Stephanomeria.
Mechanisms
According to Simpson (1944), quantum evolution resulted from Sewall Wright's model of random genetic drift. Simpson believed that major evolutionary transitions would arise when small populations, that were isolated and limited from gene flow,
would fixate upon unusual gene combinations. This "inadaptive phase"
(caused by genetic drift) would then (by natural selection) drive a deme population from one stable adaptive peak to another on the adaptive fitness landscape. However, in his Major Features of Evolution (1953) Simpson wrote that this mechanism was still controversial:
"whether prospective adaptation as prelude to quantum
evolution arises adaptively or inadaptively. It was concluded above that
it usually arises adaptively . . . . The precise role of, say, genetic
drift in this process thus is largely speculative at present. It may
have an essential part or none. It surely is not involved in all cases
of quantum evolution, but there is a strong possibility that it is often
involved. If or when it is involved, it is an initiating mechanism.
Drift can only rarely, and only for lower categories, have completed the
transition to a new adaptive zone."
This preference for adaptive over inadaptive forces led Stephen Jay Gould
to call attention to the "hardening of the Modern Synthesis", a trend
in the 1950s where adaptationism took precedence over the pluralism of
mechanisms common in the 1930s and 40s.
Simpson considered quantum evolution his crowning achievement,
being "perhaps the most important outcome of [my] investigation, but
also the most controversial and hypothetical."
Yellow chamomile head showing the Fibonacci numbers in spirals consisting of 21 (blue) and 13 (aqua). Such arrangements have been noticed since the Middle Ages and can be used to make mathematical models of a wide variety of plants.
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to test scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side.
Theoretical biology focuses more on the development of theoretical
principles for biology while mathematical biology focuses on the use of
mathematical tools to study biological systems, even though the two
terms are sometimes interchanged.
Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and practical
research. Describing systems in a quantitative manner means their
behavior can be better simulated, and hence properties can be predicted
that might not be evident to the experimenter. This requires precise mathematical models.
Because of the complexity of the living systems, theoretical biology employs several fields of mathematics, and has contributed to the development of new techniques.
History
Early history
Mathematics has been used in biology as early as the 13th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century, Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre François Verhulst formulated the logistic growth model in 1836.
Fritz Müller described the evolutionary benefits of what is now called Müllerian mimicry in 1879, in an account notable for being the first use of a mathematical argument in evolutionary ecology to show how powerful the effect of natural selection would be, unless one includes Malthus's discussion of the effects of population growth that influenced Charles Darwin: Malthus argued that growth would be exponential (he uses the word "geometric") while resources (the environment's carrying capacity) could only grow arithmetically.
Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include:
The rapid growth of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools
Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology
An increase in computing power, which facilitates calculations and simulations not previously possible
An increasing interest in in silico
experimentation due to ethical considerations, risk, unreliability and
other complications involved in human and animal research
Areas of research
Several areas of specialized research in mathematical and theoretical biologyas well as external links to related projects in various universities
are concisely presented in the following subsections, including also a
large number of appropriate validating references from a list of several
thousands of published authors contributing to this field. Many of the
included examples are characterised by highly complex, nonlinear, and
supercomplex mechanisms, as it is being increasingly recognised that the
result of such interactions may only be understood through a
combination of mathematical, logical, physical/chemical, molecular and
computational models.
Abstract relational biology
Abstract
relational biology (ARB) is concerned with the study of general,
relational models of complex biological systems, usually abstracting out
specific morphological, or anatomical, structures. Some of the simplest
models in ARB are the Metabolic-Replication, or (M,R)--systems
introduced by Robert Rosen in 1957–1958 as abstract, relational models
of cellular and organismal organization.
Other approaches include the notion of autopoiesis developed by Maturana and Varela, Kauffman's Work-Constraints cycles, and more recently the notion of closure of constraints.
An
elaboration of systems biology to understand the more complex life
processes was developed since 1970 in connection with molecular set
theory, relational biology and algebraic biology.
Evolutionary biology has been the subject of extensive
mathematical theorizing. The traditional approach in this area, which
includes complications from genetics, is population genetics. Most population geneticists consider the appearance of new alleles by mutation, the appearance of new genotypes by recombination, and changes in the frequencies of existing alleles and genotypes at a small number of geneloci. When infinitesimal effects at a large number of gene loci are considered, together with the assumption of linkage equilibrium or quasi-linkage equilibrium, one derives quantitative genetics. Ronald Fisher made fundamental advances in statistics, such as analysis of variance,
via his work on quantitative genetics. Another important branch of
population genetics that led to the extensive development of coalescent theory is phylogenetics.
Phylogenetics is an area that deals with the reconstruction and
analysis of phylogenetic (evolutionary) trees and networks based on
inherited characteristics Traditional population genetic models deal with alleles and genotypes, and are frequently stochastic.
Many population genetics models assume that population sizes are
constant. Variable population sizes, often in the absence of genetic
variation, are treated by the field of population dynamics. Work in this area dates back to the 19th century, and even as far as 1798 when Thomas Malthus formulated the first principle of population dynamics, which later became known as the Malthusian growth model. The Lotka–Volterra predator-prey equations are another famous example. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of the spread of infections have been proposed and analyzed, and provide important results that may be applied to health policy decisions.
The earlier stages of mathematical biology were dominated by mathematical biophysics,
described as the application of mathematics in biophysics, often
involving specific physical/mathematical models of biosystems and their
components or compartments.
The following is a list of mathematical descriptions and their assumptions.
Deterministic processes (dynamical systems)
A
fixed mapping between an initial state and a final state. Starting from
an initial condition and moving forward in time, a deterministic
process always generates the same trajectory, and no two trajectories
cross in state space.
A random mapping between an initial state and a final state, making the state of the system a random variable with a corresponding probability distribution.
Non-Markovian processes – generalized master equation –
continuous time with memory of past events, discrete state space,
waiting times of events (or transitions between states) discretely
occur.
A
model of a biological system is converted into a system of equations,
although the word 'model' is often used synonymously with the system of
corresponding equations. The solution of the equations, by either
analytical or numerical means, describes how the biological system
behaves either over time or at equilibrium.
There are many different types of equations and the type of behavior
that can occur is dependent on both the model and the equations used.
The model often makes assumptions about the system. The equations may
also make assumptions about the nature of what may occur.
Molecular set theory
Molecular set theory (MST) is a mathematical formulation of the wide-sense chemical kinetics
of biomolecular reactions in terms of sets of molecules and their
chemical transformations represented by set-theoretical mappings between
molecular sets. It was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in mathematical medicine.
In a more general sense, MST is the theory of molecular categories
defined as categories of molecular sets and their chemical
transformations represented as set-theoretical mappings of molecular
sets. The theory has also contributed to biostatistics and the
formulation of clinical biochemistry problems in mathematical
formulations of pathological, biochemical changes of interest to
Physiology, Clinical Biochemistry and Medicine.
Organizational biology
Theoretical
approaches to biological organization aim to understand the
interdependence between the parts of organisms. They emphasize the
circularities that these interdependences lead to. Theoretical
biologists developed several concepts to formalize this idea.
For example, abstract relational biology (ARB)
is concerned with the study of general, relational models of complex
biological systems, usually abstracting out specific morphological, or
anatomical, structures. Some of the simplest models in ARB are the
Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957–1958 as abstract, relational models of cellular and organismal organization.
The eukaryotic cell cycle is very complex and has been the subject of intense study, since its misregulation leads to cancers.
It is possibly a good example of a mathematical model as it deals with
simple calculus but gives valid results. Two research groups
have produced several models of the cell cycle simulating several
organisms. They have recently produced a generic eukaryotic cell cycle
model that can represent a particular eukaryote depending on the values
of the parameters, demonstrating that the idiosyncrasies of the
individual cell cycles are due to different protein concentrations and
affinities, while the underlying mechanisms are conserved (Csikasz-Nagy
et al., 2006).
By means of a system of ordinary differential equations these models show the change in time (dynamical system) of the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process).
To obtain these equations an iterative series of steps must be
done: first the several models and observations are combined to form a
consensus diagram and the appropriate kinetic laws are chosen to write
the differential equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate reactions and Goldbeter–Koshland kinetics
for ultrasensitive transcription factors, afterwards the parameters of
the equations (rate constants, enzyme efficiency coefficients and
Michaelis constants) must be fitted to match observations; when they
cannot be fitted the kinetic equation is revised and when that is not
possible the wiring diagram is modified. The parameters are fitted and
validated using observations of both wild type and mutants, such as
protein half-life and cell size.
To fit the parameters, the differential equations must be
studied. This can be done either by simulation or by analysis. In a
simulation, given a starting vector
(list of the values of the variables), the progression of the system is
calculated by solving the equations at each time-frame in small
increments.
In analysis, the properties of the equations are used to investigate
the behavior of the system depending on the values of the parameters and
variables. A system of differential equations can be represented as a vector field,
where each vector described the change (in concentration of two or more
protein) determining where and how fast the trajectory (simulation) is
heading. Vector fields can have several special points: a stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an unstable point, either a source or a saddle point,
which repels (forcing the concentrations to change away from a certain
value), and a limit cycle, a closed trajectory towards which several
trajectories spiral towards (making the concentrations oscillate).
A better representation, which handles the large number of variables and parameters, is a bifurcation diagram using bifurcation theory.
The presence of these special steady-state points at certain values of a
parameter (e.g. mass) is represented by a point and once the parameter
passes a certain value, a qualitative change occurs, called a
bifurcation, in which the nature of the space changes, with profound
consequences for the protein concentrations: the cell cycle has phases
(partially corresponding to G1 and G2) in which mass, via a stable
point, controls cyclin levels, and phases (S and M phases) in which the
concentrations change independently, but once the phase has changed at a
bifurcation event (Cell cycle checkpoint),
the system cannot go back to the previous levels since at the current
mass the vector field is profoundly different and the mass cannot be
reversed back through the bifurcation event, making a checkpoint
irreversible. In particular the S and M checkpoints are regulated by
means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation.
Charles Darwin's On the Origin of Species
was published in 1859, arguing from circumstantial evidence that
selection by human breeders could produce change, and that since there
was clearly a struggle for existence, that natural selection must be
taking place. But he lacked an explanation either for genetic variation or for heredity, both essential to the theory. Many alternative theories were accordingly considered by biologists, threatening to undermine Darwinian evolution.
Some of the first evidence was provided by Darwin's contemporaries, the naturalists Henry Walter Bates and Fritz Müller.
They described forms of mimicry that now carry their names, based on
their observations of tropical butterflies. These highly specific
patterns of coloration are readily explained by natural selection, since
predators such as birds which hunt by sight will more often catch and
kill insects that are less good mimics of distasteful models than those
that are better mimics, but the patterns are otherwise hard to explain.
Darwinists such as Alfred Russel Wallace and Edward Bagnall Poulton, and in the 20th century Hugh Cott and Bernard Kettlewell, sought evidence that natural selection was taking place. Wallace noted that snow camouflage,
especially plumage and pelage that changed with the seasons, suggested
an obvious explanation as an adaptation for concealment. Poulton's 1890
book, The Colours of Animals, written during Darwinism's lowest ebb,
used all the forms of coloration to argue the case for natural
selection. Cott described many kinds of camouflage, and in particular
his drawings of coincident disruptive coloration
in frogs convinced other biologists that these deceptive markings were
products of natural selection. Kettlewell experimented on peppered moth evolution, showing that the species had adapted as pollution changed the environment; this provided compelling evidence of Darwinian evolution.
Context
Charles Darwin published On the Origin of Species in 1859, arguing that evolution in nature must be driven by natural selection, just as breeds of domestic animals and cultivars of crop plants were driven by artificial selection. Darwin's theory radically altered popular and scientific opinion about the development of life.
However, he lacked evidence and explanations for some critical
components of the evolutionary process. He could not explain the source
of variation in traits within a species, and did not have a mechanism of heredity that could pass traits faithfully from one generation to the next. This made his theory vulnerable; alternative theories were being explored during the eclipse of Darwinism;
and so Darwinian field naturalists like Wallace, Bates and Müller
looked for clear evidence that natural selection actually occurred.
Animal coloration, readily observable, soon provided strong and
independent lines of evidence, from camouflage, mimicry and aposematism,
that natural selection was indeed at work. The historian of science Peter J. Bowler wrote that Darwin's theory "was also extended to the broader topics of protective resemblances and mimicry, and this was its greatest triumph in explaining adaptations".
Convergent evolution of snow camouflage in Arctic hare, ermine, and ptarmigan provided early evidence for natural selection.
In his 1889 book Darwinism, the naturalist Alfred Russel Wallace considered the white coloration of Arctic animals. He recorded that the Arctic fox, Arctic hare, ermine and ptarmigan change their colour seasonally, and gave "the obvious explanation", that it was for concealment.
The modern ornithologist W. L. N. Tickell, reviewing proposed
explanations of white plumage in birds, writes that in the ptarmigan "it
is difficult to escape the conclusion that cryptic
brown summer plumage becomes a liability in snow, and white plumage is
therefore another cryptic adaptation." All the same, he notes, "in spite
of winter plumage, many Ptarmigan in NE Iceland are killed by Gyrfalcons throughout the winter."
More recently, decreasing snow cover in Poland, caused by global warming, is reflected in a reduced percentage of white-coated weasels
that become white in winter. Days with snow cover halved between 1997
and 2007, and as few as 20 percent of the weasels had white winter
coats. This was shown to be a result of natural selection by predators
making use of camouflage mismatch.
Coincident disruptive coloration
Hugh Cott's drawings of 'coincident disruptive coloration' formed "persuasive arguments" for natural selection. Left: active; right: at rest, marks coinciding.
In particular, they argue, "Coincident Disruptive Coloration"
(one of Cott's categories) "made Cott's drawings the most compelling
evidence for natural selection enhancing survival through disruptive camouflage."
Cott explained, while discussing "a little frog known as Megalixalus fornasinii"
in his chapter on coincident disruptive coloration, that "it is only
when the pattern is considered in relation to the frog's normal attitude
of rest that its remarkable nature becomes apparent... The attitude and
very striking colour-scheme thus combine to produce an extraordinary
effect, whose deceptive appearance depends upon the breaking up of the
entire form into two strongly contrasted areas of brown and white.
Considered separately, neither part resembles part of a frog. Together
in nature the white configuration alone is conspicuous. This stands out
and distracts the observer's attention from the true form and contour of
the body and appendages on which it is superimposed".
Cott concluded that the effect was concealment "so long as the false configuration is recognized in preference to the real one".
Such patterns embody, as Cott stressed, considerable precision as the
markings must line up accurately for the disguise to work. Cott's
description and in particular his drawings convinced biologists that the
markings, and hence the camouflage, must have survival value (rather
than occurring by chance); and further, as Cuthill and Székely indicate,
that the bodies of animals that have such patterns must indeed have
been shaped by natural selection.
Bernard Kettlewell claimed that changes in the frequencies of light and dark morphs of the peppered moth, Biston betularia were direct evidence of natural selection.
Between 1953 and 1956, the geneticist Bernard Kettlewell experimented on peppered moth evolution.
He presented results showing that in a polluted urban wood with dark
tree trunks, dark moths survived better than pale ones, causing industrial melanism,
whereas in a clean rural wood with paler trunks, pale moths survived
better than dark ones. The implication was that survival was caused by
camouflage against suitable backgrounds, where predators hunting by
sight (insect-eating birds, such as the great tits
used in the experiment) selectively caught and killed the less
well-camouflaged moths. The results were intensely controversial, and
from 2001 Michael Majerus
carefully repeated the experiment. The results were published
posthumously in 2012, vindicating Kettlewell's work as "the most direct
evidence", and "one of the clearest and most easily understood examples
of Darwinian evolution in action".
Batesian mimicry, named for the 19th century naturalist Henry Walter Bates who first noted the effect in 1861, "provides numerous excellent examples of natural selection" at work. The evolutionary entomologist James Mallet noted that mimicry was "arguably the oldest Darwinian theory not attributable to Darwin." Inspired by On the Origin of Species,
Bates realized that unrelated Amazonian butterflies resembled each
other when they lived in the same areas, but had different coloration in
different locations in the Amazon, something that could only have been
caused by adaptation.
Müllerian mimicry,
too, in which two or more distasteful species that share one or more
predators have come to mimic each other's warning signals, was clearly
adaptive; Fritz Müller described the effect in 1879, in an account notable for being the first use of a mathematical argument in evolutionary ecology to show how powerful the effect of natural selection would be.
In 1867, in a letter to Darwin, Wallace described warning coloration. The evolutionary zoologist James Mallet notes that this discovery "rather illogically"
followed rather than preceded the accounts of Batesian and Müllerian
mimicry, which both rely on the existence and effectiveness of warning
coloration. The conspicuous colours and patterns of animals with strong defences such as toxins are advertised to predators, signalling honestly that the animal is not worth attacking. This directly increases the reproductive fitness
of the potential prey, providing a strong selective advantage. The
existence of unequivocal warning coloration is therefore clear evidence
of natural selection at work.
Edward Bagnall Poulton's 1890 book, The Colours of Animals,
renamed Wallace's concept of warning colours "aposematic" coloration,
as well as supporting Darwin's then unpopular theories of natural
selection and sexual selection. Poulton's explanations of coloration are emphatically Darwinian. For example, on aposematic coloration he wrote that
At first sight the existence of
this group seems to be a difficulty in the way of the general
applicability of the theory of natural selection. Warning Colours appear
to benefit the would-be enemies rather than the conspicuous forms
themselves, and the origin and growth of a character intended solely for
the advantage of some other species cannot be explained by the theory
of natural selection. But the conspicuous animal is greatly benefited by
its Warning Colours. If it resembled its surroundings like the members
of the other class, it would be liable to a great deal of accidental or
experimental tasting, and there would be nothing about it to impress the
memory of an enemy, and thus to prevent the continual destruction of
individuals. The object of Warning Colours is to assist the education of
enemies, enabling them to easily learn and remember the animals which
are to be avoided. The great advantage conferred upon the conspicuous
species is obvious when it is remembered that such an easy and
successful education means an education involving only a small sacrifice
of life."
Poulton summed up his allegiance to Darwinism as an explanation of
Batesian mimicry in one sentence: "Every step in the gradually
increasing change of the mimicking in the direction of specially
protected form, would have been an advantage in the struggle for
existence".
The historian of biology Peter J. Bowler
commented that Poulton used his book to complain about
experimentalists' lack of attention to what field naturalists (like
Wallace, Bates, and Poulton) could readily see were adaptive features.
Bowler added that "The fact that the adaptive significance of coloration
was (sic) widely challenged indicates just how far anti-Darwinian feeling had developed.
Only field naturalists such as Poulton refused to give in, convinced
that their observations showed the validity of selection, whatever the
theoretical problems."
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