6,908 research outputs found
How to read probability distributions as statements about process
Probability distributions can be read as simple expressions of information.
Each continuous probability distribution describes how information changes with
magnitude. Once one learns to read a probability distribution as a measurement
scale of information, opportunities arise to understand the processes that
generate the commonly observed patterns. Probability expressions may be parsed
into four components: the dissipation of all information, except the
preservation of average values, taken over the measurement scale that relates
changes in observed values to changes in information, and the transformation
from the underlying scale on which information dissipates to alternative scales
on which probability pattern may be expressed. Information invariances set the
commonly observed measurement scales and the relations between them. In
particular, a measurement scale for information is defined by its invariance to
specific transformations of underlying values into measurable outputs.
Essentially all common distributions can be understood within this simple
framework of information invariance and measurement scale.Comment: v2: added table of contents, adjusted section numbers v3: minor
editing, updated referenc
Receptor uptake arrays for vitamin B12, siderophores and glycans shape bacterial communities
Molecular variants of vitamin B12, siderophores and glycans occur. To take up
variant forms, bacteria may express an array of receptors. The gut microbe
Bacteroides thetaiotaomicron has three different receptors to take up variants
of vitamin B12 and 88 receptors to take up various glycans. The design of
receptor arrays reflects key processes that shape cellular evolution.
Competition may focus each species on a subset of the available nutrient
diversity. Some gut bacteria can take up only a narrow range of carbohydrates,
whereas species such as B.~thetaiotaomicron can digest many different complex
glycans. Comparison of different nutrients, habitats, and genomes provide
opportunity to test hypotheses about the breadth of receptor arrays. Another
important process concerns fluctuations in nutrient availability. Such
fluctuations enhance the value of cellular sensors, which gain information
about environmental availability and adjust receptor deployment. Bacteria often
adjust receptor expression in response to fluctuations of particular
carbohydrate food sources. Some species may adjust expression of uptake
receptors for specific siderophores. How do cells use sensor information to
control the response to fluctuations? That question about regulatory wiring
relates to problems that arise in control theory and artificial intelligence.
Control theory clarifies how to analyze environmental fluctuations in relation
to the design of sensors and response systems. Recent advances in deep learning
studies of artificial intelligence focus on the architecture of regulatory
wiring and the ways in which complex control networks represent and classify
environmental states. I emphasize the similar design problems that arise in
cellular evolution, control theory, and artificial intelligence. I connect
those broad concepts to testable hypotheses for bacterial uptake of B12,
siderophores and glycans.Comment: Added many new references, edited throughou
Microbial metabolism: optimal control of uptake versus synthesis
Microbes require several complex organic molecules for growth. A species may
obtain a required factor by taking up molecules released by other species or by
synthesizing the molecule. The patterns of uptake and synthesis set a flow of
resources through the multiple species that create a microbial community. This
article analyzes a simple mathematical model of the tradeoff between uptake and
synthesis. Key factors include the influx rate from external sources relative
to the outflux rate, the rate of internal decay within cells, and the cost of
synthesis. Aspects of demography also matter, such as cellular birth and death
rates, the expected time course of a local resource flow, and the associated
lifespan of the local population. Spatial patterns of genetic variability and
differentiation between populations may also strongly influence the evolution
of metabolic regulatory controls of individual species and thus the structuring
of microbial communities. The widespread use of optimality approaches in recent
work on microbial metabolism has ignored demography and genetic structure
The Price equation program: simple invariances unify population dynamics, thermodynamics, probability, information and inference
The fundamental equations of various disciplines often seem to share the same
basic structure. Natural selection increases information in the same way that
Bayesian updating increases information. Thermodynamics and the forms of common
probability distributions express maximum increase in entropy, which appears
mathematically as loss of information. Physical mechanics follows paths of
change that maximize Fisher information. The information expressions typically
have analogous interpretations as the Newtonian balance between force and
acceleration, representing a partition between direct causes of change and
opposing changes in the frame of reference. This web of vague analogies hints
at a deeper common mathematical structure. I suggest that the Price equation
expresses that underlying universal structure. The abstract Price equation
describes dynamics as the change between two sets. One component of dynamics
expresses the change in the frequency of things, holding constant the values
associated with things. The other component of dynamics expresses the change in
the values of things, holding constant the frequency of things. The separation
of frequency from value generalizes Shannon's separation of the frequency of
symbols from the meaning of symbols in information theory. The Price equation's
generalized separation of frequency and value reveals a few simple invariances
that define universal geometric aspects of change. For example, the
conservation of total frequency, although a trivial invariance by itself,
creates a powerful constraint on the geometry of change. That constraint plus a
few others seem to explain the common structural forms of the equations in
different disciplines. From that abstract perspective, interpretations such as
selection, information, entropy, force, acceleration, and physical work arise
from the same underlying geometry expressed by the Price equation.Comment: Version 3: added figure illustrating geometry; added table of symbols
and two tables summarizing mathematical relations; this version accepted for
publication in Entrop
The invariances of power law size distributions
Size varies. Small things are typically more frequent than large things. The
logarithm of frequency often declines linearly with the logarithm of size. That
power law relation forms one of the common patterns of nature. Why does the
complexity of nature reduce to such a simple pattern? Why do things as
different as tree size and enzyme rate follow similarly simple patterns? Here I
analyze such patterns by their invariant properties. For example, a common
pattern should not change when adding a constant value to all observations.
That shift is essentially the renumbering of the points on a ruler without
changing the metric information provided by the ruler. A ruler is shift
invariant only when its scale is properly calibrated to the pattern being
measured. Stretch invariance corresponds to the conservation of the total
amount of something, such as the total biomass and consequently the average
size. Rotational invariance corresponds to pattern that does not depend on the
order in which underlying processes occur, for example, a scale that additively
combines the component processes leading to observed values. I use tree size as
an example to illustrate how the key invariances shape pattern. A simple
interpretation of common pattern follows. That simple interpretation connects
the normal distribution to a wide variety of other common patterns through the
transformations of scale set by the fundamental invariances.Comment: Added appendix discussing the lognormal distribution, updated to
match version 2 of published version at F1000Researc
Input-output relations in biological systems: measurement, information and the Hill equation
Biological systems produce outputs in response to variable inputs.
Input-output relations tend to follow a few regular patterns. For example, many
chemical processes follow the S-shaped Hill equation relation between input
concentrations and output concentrations. That Hill equation pattern
contradicts the fundamental Michaelis-Menten theory of enzyme kinetics. I use
the discrepancy between the expected Michaelis-Menten process of enzyme
kinetics and the widely observed Hill equation pattern of biological systems to
explore the general properties of biological input-output relations. I start
with the various processes that could explain the discrepancy between basic
chemistry and biological pattern. I then expand the analysis to consider
broader aspects that shape biological input-output relations. Key aspects
include the input-output processing by component subsystems and how those
components combine to determine the system's overall input-output relations.
That aggregate structure often imposes strong regularity on underlying
disorder. Aggregation imposes order by dissipating information as it flows
through the components of a system. The dissipation of information may be
evaluated by the analysis of measurement and precision, explaining why certain
common scaling patterns arise so frequently in input-output relations. I
discuss how aggregation, measurement and scale provide a framework for
understanding the relations between pattern and process. The regularity imposed
by those broader structural aspects sets the contours of variation in biology.
Thus, biological design will also tend to follow those contours. Natural
selection may act primarily to modulate system properties within those broad
constraints.Comment: Biology Direct 8:3
Generative models versus underlying symmetries to explain biological pattern
Mathematical models play an increasingly important role in the interpretation
of biological experiments. Studies often present a model that generates the
observations, connecting hypothesized process to an observed pattern. Such
generative models confirm the plausibility of an explanation and make testable
hypotheses for further experiments. However, studies rarely consider the broad
family of alternative models that match the same observed pattern. The
symmetries that define the broad class of matching models are in fact the only
aspects of information truly revealed by observed pattern. Commonly observed
patterns derive from simple underlying symmetries. This article illustrates the
problem by showing the symmetry associated with the observed rate of increase
in fitness in a constant environment. That underlying symmetry reveals how each
particular generative model defines a single example within the broad class of
matching models. Further progress on the relation between pattern and process
requires deeper consideration of the underlying symmetries
Simple unity among the fundamental equations of science
The Price equation describes the change in populations. Change concerns some
value, such as biological fitness, information or physical work. The Price
equation reveals universal aspects for the nature of change, independently of
the meaning ascribed to values. By understanding those universal aspects, we
can see more clearly why fundamental mathematical results in different
disciplines often share a common form. We can also interpret more clearly the
meaning of key results within each discipline. For example, the mathematics of
natural selection in biology has a form closely related to information theory
and physical entropy. Does that mean that natural selection is about
information or entropy? Or do natural selection, information and entropy arise
as interpretations of a common underlying abstraction? The Price equation
suggests the latter. The Price equation achieves its abstract generality by
partitioning change into two terms. The first term naturally associates with
the direct forces that cause change. The second term naturally associates with
the changing frame of reference. In the Price equation's canonical form, total
change remains zero because the conservation of total probability requires that
all probabilities invariantly sum to one. Much of the shared common form for
the mathematics of different disciplines may arise from that seemingly trivial
invariance of total probability, which leads to the partitioning of total
change into equal and opposite components of the direct forces and the changing
frame of reference.Comment: arXiv admin note: text overlap with arXiv:1810.0926
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