A framework for parsing heritable information
Antony M Jose111correspondence: email -<EMAIL_ADDRESS>mail - Rm 2136,
Bioscience Research Building, 4066 Campus Drive, University of Maryland,
College Park, MD - 20742, USA.
Department of Cell Biology and Molecular Genetics,
University of Maryland,
College Park, MD 20742, USA

Living systems transmit heritable information using the replicating gene
sequences and the cycling regulators assembled around gene sequences. Here I
develop a framework for heredity and development that includes the cycling
regulators parsed in terms of what an organism can sense about itself and its
environment by defining entities, their sensors, and the sensed properties.
Entities include small molecules (ATP, ions, metabolites, etc.),
macromolecules (individual proteins, RNAs, polysaccharides, etc.), and
assemblies of molecules. While concentration may be the only relevant property
measured by sensors for small molecules, multiple properties that include
concentration, sequence, conformation, and modification may all be measured
for macromolecules and assemblies. Each configuration of these entities and
sensors that is recreated in successive generations in a given environment
thus specifies a potentially vast amount of information driving complex
development in each generation. This Entity-Sensor-Property framework explains
how sensors limit the number of distinguishable states, how distinct molecular
configurations can be functionally equivalent, and how regulation of sensors
prevents detection of some perturbations. Overall, this framework is a useful
guide for understanding how life evolves and how the storage of information
has itself evolved with complexity since before the origin of life.

Keywords: systems biology, information theory, homeostasis, transgenerational
inheritance, synthetic biology

Introduction

Analyses of living systems from molecular to population scales have revealed
information storage and processing across multiple scales as key attributes of
life [1]. The need to understand the behavior of a basic unit of life - a
single cell - in terms of an integrated framework for information handling has
been previously articulated [2, 3, 4, 5], but is yet to be developed. A single
cell is often the bottleneck stage that separates successive generations,
making it the minimal space for storing all heritable information (see
supplemental text for variations on the single-cell bottleneck). Such
information in molecules are part of the ‘nature’ of organisms and do not
include information transmitted when parents train progeny, which can be
considered as ‘nurture’. Cells and more complex living systems can change
their information content by learning through interactions with their
environment. However, their ability to transmit any such learned information
from one generation to the next is limited by the available storage in the
bottleneck stage and potentially other system constraints (e.g. inability of
learned information to cross generational boundaries)[6]. To appreciate these
limits, we need to consider the total amount of information that could be
encoded using all molecules in the bottleneck stage. Such joint consideration
of all heritable information that is transmissible using molecules will inform
how complexity grows over evolutionary time, what constitutes nature versus
nurture, and how to synthesize new living systems.

To facilitate discussion of all heritable information, I begin by defining key
terms introduced in an earlier article [6]: stores of information, stored
information, and cell code. Stores(n.) of information refer to molecules or
arrangements of molecules that hold information. This information can be
transferred to other molecules or arrangements and the original store can be
degraded or modified after such transfer. Therefore, molecules and
arrangements of molecules can have stored(v.) information. Cell code(n.)
refers to the heritable information that encodes the development of organisms
from a bottleneck stage, which is minimally a single cell. Similar development
in successive generations in a given environment presumably relies on similar
cell codes assembled during bottleneck stages (see supplemental text for more
on assembly of cell codes).

The information in a cell code can be conceptually separated into two distinct
forms [6]. One is the genome sequence, where information is stored in a linear
sequence of bases, and the other is the recurring arrangement, where
information is stored in the concentrations, configurations, and interactions
of molecules in bottleneck stages (see supplemental text on cell code
assembly). While the information content in this arrangement and the extent to
which it is recreated is currently not easily quantified, it is clear that
heredity relies on information that is held in multiple stores and transmitted
across generations. This communication of heritable information through the
development of an organism from one generation to the next has been likened to
the transmission of messages through a communication channel from sender to
receiver (e.g. refs. [7, 6]). Just as ‘the fundamental problem of
communication is that of reproducing at one point either exactly or
approximately a message selected at another point’ [8], the fundamental
problem of heredity is that of reproducing at one bottleneck stage either
exactly or approximately a cell code selected at the preceding bottleneck
stage.

Each choice of molecules and their arrangement in bottleneck stages
collectively stores heritable information and forms a message transmitted
across generations. The average information content in a message chosen from
among N possible messages is given by the Shannon entropy,
$H=\sum_{i=1}^{N}-p_{i}\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}log_{2}(p_{i})$,
where $p_{i}$ is the probability of the $i^{th}$ message [8, 9]. If the
probability of selecting each message is equal, this expression simplifies to
give the maximal information a message can carry,
$H=\sum_{i=1}^{N}-1/N\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}log_{2}(1/N)=log_{2}N$
bits. Therefore, to determine the maximal heritable information in a living
system, we need to enumerate all distinguishable states of its bottleneck
stage (i.e., $N$). This exercise will provide a starting point for the joint
analysis of all heritable information that needs to be transmitted across
generational boundaries for the reproduction of living systems.

Static and dynamic storage of information

The logical requirements for self-replication have been explored in two-
dimensional universes called cellular automata using abstract ‘machines’
[10]222Individual units in these machines are called ‘cells’, but are referred
to as ‘parts’ in this article to avoid confusion with biological cells.. Of
particular relevance are self-replicating machines that use the same store of
information in two distinct ways: (1) as instructions whose interpretation
leads to the construction of an identical copy of the machine, and (2) as data
to be copied without interpretation and placed in the copied machine.

Figure 1: Figure 1. Self-replicating ‘machines’ with instructions held in a
static tape or in a dynamic tape have been implemented in cellular automata.
(A) Implementation of John von Neumann’s design of universal constructor [11].
Top, The universal constructor in the starting configuration. Inset, Schematic
of broad regions within the universal constructor. Bottom, The 32 states of
the parts that make up the machine (see [11] for the meaning of each color).
(B) Implementation of the Langton loop [12]. Left, The loop in the starting
configuration. Middle, A replication intermediate showing use of all states.
Right, Loops near the end of one round of replication. Bottom, The 8 states
used for replicating the loop (see [12] for the meaning of each color). Red
bar indicates scale for comparing (A) and (B). See methods for additional
details.

This scheme for making self-replicating machines avoids the infinite regress
of instructions stored within instructions and is often presented as analogous
to the process of self-replication in living systems with the ‘instructions’
being held in DNA. However, the instructions for replicating a machine can be
held either in static tapes (e.g., the von Neumann universal constructor,
Figure 1A) or in dynamic tapes (e.g., the Langton loop, Figure 1B).

These two different types of instruction storage can be viewed as occuring
simultaneously in living systems as the ‘static’ genome sequence and the
‘dynamic’ recurring arrangement of molecules [6].

The transmission of the genome sequence from one generation to the next occurs
along a lineage of cells that each go through cell division cycles. As a
result, numerous additional cycling stores can carry information across
generations. For example, the information for copying a genome is stored in an
arrangement of molecules that changes during replication such that the genome
is usually copied with a period of one cell division cycle. Additional cycling
stores of information are clearly recognizable in oscillations that occur at
different temporal and spatial scales relative to generation time. These
include oscillations in the chemical modifications of molecules (e.g., a
$\sim$24 hr period in cyanobacteria [13] despite a $\sim$12 hr cell cycle
[14]), in the localizations of molecules (e.g., a $\sim$40 s period in cell
lines that have a $\sim$24 hr cell cycle [15]), in the collective morphology
of embryonic cells (e.g., a $\sim$0.5 hr period in sea anemone that have a
$\sim$0.5 hr cell cycle [16]), and in the activity levels of circuits (e.g.,
$\sim$24 hr circadian rhythms in non-cycling neurons [17]).

From these considerations, the following realizations emerge about living
systems:

(1) The transmission of form and function across generations relies on many
stores of information that cycle with different periods that could each in
principle range from less than the duration of one cell division cycle to more
than that of one generation.

(2) The relative phases of the many cycles within the continuous lineage of
cells between generations creates distinct states over time such that the cell
code for the development of an organism is approximated at the start of each
generation.

Thus, the integrated process of self-replication cannot be artificially parsed
into the static genome that holds all the instructions to be interpreted by
the dynamic molecular machines in the cell.

Information in self-replicating machines

Consideration of the total information stored in a self-replicating machine
can clarify the different stores of information required for replication and
sharpen the corresponding unknowns in living systems. For example, consider
the self-replicating universal constructor (Figure 1A), which has a ‘machine’
that has 6,329 parts with 32 states per part and uses an instruction tape that
has 145,315 parts with 2 states per part. The maximal information stored in
this machine could be enumerated by separately considering three different
stores that each have analogs in living systems: (1) the configuration or
shape of the machine, (2) the instruction tape, and (3) the parts of the
machine.

The information stored in the shape of the machine is incalculably large
because we have to consider the universe of shapes from which the particular
assembly of parts that make the machine was selected (see supplemental text
and Supplemental Figure 1 for a proof). This information is akin to the
information required for getting together the particular collection of
molecules that constitutes each current living system and has accrued since
before the origin of life along the lineage of every living system. Because
the unknown information in all historical environments (i.e. past available
complements of molecules) needs to be taken into account to determine what
life accrued bit by bit [18], the magnitude of this information is
incalculable.

The maximal information that can be stored in the instruction tape that has
$N=145,315$ parts with two states each is given by $H=log_{2}2^{N}=N=145,315$
bits. This store is analogous to the linear genome where the information is
stored in the sequence of the four bases in DNA (A, T, G, C). For such a
genome of length $L$, $H=log_{2}4^{L}=2L$ bits.

The maximal information that can be stored in the machine that has $N=6,329$
parts with 32 states each is given by $H=log_{2}32^{N}=5N=31,645$ bits. This
store could be analogous to everything other than the genome sequence within
the bottleneck stage. However, unlike in cellular automata, discrete states of
living systems are not easily defined. Calculating the information in this
potentially vast store requires definition of the biologically relevant states
of the bottleneck stage. For any given genome, knowing the rest of the cell
code for different organisms is a prerequisite for constructing living systems
of varying complexity.

Here, I develop a framework for all the heritable information in a living
system in terms of what that system can sense about itself and its
environment. This framework is useful for guiding the experimental analysis of
living systems and potentially for the design and analysis of other persistent
adapting systems.

Heritable information in living systems

The spatial arrangement of the genome and everything else within the
bottleneck stage could change over the course of development such that similar
arrangements are reached with a period of one life cycle. As a result,
molecules that are part of the recurring cell code could play different roles
throughout development and defy permenant classification based on their roles.
For example, an abundant maternal RNA that is simply used as a source of
nucleotides in the developing embryo could at a later stage become a message
that is translated into a protein. Nevertheless, a temporary classification
during the bottleneck stage is necessary to enumerate the bits of information
stored in cell codes. To facilitate this enumeration in units that are
relevant for each living system and its environment, I propose considering
entities, their sensors, and the sensed properties.

Entities. An entity is a molecule333For simplicity, the term molecule is used
to refer to everything found in a living system that is chemically isolatable
such as ions, atoms, and chemically bonded collections of atoms, and is
extensible to all factors that remain to be discovered. or association of
molecules within a living system or in the environment that interacts with the
living system. A cell code can include entities that are measured through
interaction with other entities sometime during the life cycle and also
entities that are never measured, which can be considered as byproducts made
by the processes of life. Such effectively inert and unmeasured entities could
nevertheless non-specifically contribute to molecular crowding at the
bottleneck stage and thereby affect interactions among other entities. While
the number of all molecules in a cell is large but countable, the
combinatorial associations of molecules could make the total number of
effective entities ($N$) larger still. Cellular components that are entities
or parts of entities include small molecules such as ATP, water, ions,
metabolites, etc., for which perhaps only concentrations are discerned by
sensors, and macromolecules such as individual proteins, RNAs,
polysaccharides, etc., for which concentrations, sequences, and conformations
may all be discerned by sensors.

Sensors. A sensor is an entity or an association of entities within the living
system that responds to changes in other entities with changes in its
properties such that these changes can result in subsequent changes in the
rest of the living system or its environment. A sensor could sense entities
within the system ($N$ total) or in the outside environment ($O$ total) that
interacts with the system (e.g., salts, nutrients, etc.). An entity that binds
or collides with another entity without any specific downstream consequences
is not considered a sensor (e.g. one water molecule bumping onto a membrane).
An entity could be a part of multiple sensors. For example, a protein complex
formed by the association of A, B, and C proteins could be detecting and
responding to the concentration of ATP while another protein complex made of A
and C could be detecting GTP. Conversely, multiple sensors could be measuring
the same entity. For example, the many kinases in the cytosol are all
potentially sensitive to the levels of a common pool of ATP. By these
definitions, ATP itself can be a sensor because its levels change in response
to production by a synthase and this change is communicated to the kinases
that respond to changes in ATP levels. All sensors are entities, but not all
entities are sensors.

Properties. A property is an attribute of an entity that is relevant for a
living system because a sensor exists that can respond to changes in the
values of that attribute. The number of different values for a property of an
entity depends on the sensor and on the regulatory constraints of the system.
Consider two sensors that can detect changes in the number of molecules of a
particular RNA: a protein Lo responds when the numbers increase by 10 and a
protein Hi responds when the numbers increase by 100. These two proteins would
thus each ‘see’ different numbers of measurable units for the same property
(number of molecules) of the same entity. However, not all detectable values
for a relevant property of an entity could be attainable because of the
regulatory constraints of the system. For example, if the RNA accumulated in
steps of 50 molecules at a time, then many of the values measurable by Lo are
never available in the living system because the system changes in steps that
are larger than the measuring step of the Lo sensor.

Environment. Organisms develop as open systems interacting with the
environment. Therefore, some entities in the environment are measured and
reacted to by the living system throughout development. Even for a constant
environment, some entities may be measured by different sensors that are
active at different times during development. As a result, living systems are
really system-environment combinations. Aspects of ‘sensing’ considered for
interactions within the system are also relevant for interactions with the
environment. Specifically, the sensed attributes of entities in the
environment depend on the nature of the evolved sensors in the system and
molecular crowders in the environment can modify the interaction between
sensors in the system and entities in the environment.

Forces. Living systems can generate and be exposed to many kinds of forces,
which are not being explicitly considered in the framework for heritable
information developed here. Rather, they are implicitly accounted for in the
properties of entities. For example, an entity experiences/exerts gravity
because of its mass, electrostatic attraction because of its charge, tension
because of its elasticity, etc. Thus, heritable information captured using
entities, sensors, and properties can account for relevant forces in the
living system or in the environment.

Configurations. The number of ways in which molecules can be arranged in the
bottleneck stage such that they can be distinguished by the system provides an
upper bound for the information that can be stored in a cell code, which is
the subset of configurations that are nearly reproduced during the bottleneck
stage of successive generations. The maximal number of such distinguishable
configurations of a living system for a given number of interacting entities
in the environment is given by the product of the number of possible genome
sequences and the number of possible systems and their coupled environments
that can support each genome sequence. Assuming that each system-environment
combination generates one characteristic set of unmeasured entities that
contribute to crowding effects, the number of distinguisable configurations
for a living system and its environment during the bottleneck stage
($C_{tot}$) is given by:

$\displaystyle C_{tot}$ $\displaystyle=\text{genomes }\times\text{ system-
environments}$ $\displaystyle C_{tot}$
$\displaystyle=X^{L}(\sum_{i=1}^{B}E_{i}(\sum_{j=1}^{S_{i}}S_{j}(\sum_{k=1}^{P_{j}}P_{k})))\hskip
30.0pt(1)$ $\displaystyle X$ $\displaystyle=\text{number of types of bases in
the genome.}$ $\displaystyle L$ $\displaystyle=\text{length of the genome in
base pairs.}$ $\displaystyle E$ $\displaystyle=\text{measured entity (total
}B\text{ in the bottleneck}$ $\displaystyle\text{\hskip 15.0pt stage:
}N_{b}\text{ in system, }O_{b}\text{ in environment).}$ $\displaystyle S$
$\displaystyle=\text{measuring sensor (total = }S_{i}\text{ for }i^{th}\text{
entity).}$ $\displaystyle=f(Y)\text{, where each
}Y\subseteq\\{E_{1},E_{2},...E_{N}\\},\text{i.e. a}$
$\displaystyle\text{\hskip 15.0ptconfiguration of entities, }N\text{ per life
cycle.}$ $\displaystyle P$ $\displaystyle=\text{attainable and measurable
values of property}$ $\displaystyle\text{\hskip 15.0pt(total = }P_{j}\text{
for }j^{th}\text{ sensor of }i^{th}\text{ entity).}$

This Entity-Sensor-Property framework enumerates all distinguishable
configurations as a product of four terms that encapsulate the maximal numbers
of distinct states in two stores of information: $X^{L}$ enumerates all
possible genome sequences, which are replicating stores of heritable
information, and $\sum_{i}E_{i}\sum_{j}S_{j}\sum_{k}P_{k}$ enumerates all
potentially recurring arrangements of interacting molecules, which are cycling
stores of heritable information. Such enumeration without considering
rearrangements of chemical bonds within any molecule can be thought of as
biological entropy and is less than the chemical entropy of an organism, which
was initially estimated allowing for rearrangements of chemical bonds to be
$\approx 4.2\times 10^{10}$ bits for E. coli [19].

It is clear that the replicating store cannot uniquely predict the cycling
store as evidenced by most distinguishable cell types of the human body all
having the same genome sequence. However, interdependence of the two stores
and compatibility with the perpetuation of life reduces this maximal number of
distinguishable states of the bottleneck stage. In other words, fewer
configurations can act as heritable cell codes ($Cell\hskip
3.0ptCode_{tot}<C_{tot}$) because of mutual constraints between the
arrangement of molecules and the genome sequence in living systems. First,
some genome sequences may not be sufficiently complex to support any living
system (e.g. a genome of all As, all Gs, all Cs, or all Ts.). Second, each
genome sequence constrains but does not dictate the number and kinds of
entities that could be part of any cell that contains the genome (e.g. DNA
sequence constrains RNA sequence, which constrains protein sequence). Third,
the genome sequence may also constrain the total number of possible
arrangements of molecules within any cell - i.e., the number of cell states
and cell types - in a given environment. Fourth, the lineage of cells that
connects two generations may be incapable of supporting some cell types
because of the need to return to the cell code at the start of each generation
within the context of a living system (i.e., some differentiated cell types
may be irreversible within the context of the living system although many can
be transformed into pluripotent stem cells in vitro [20]). The number of all
possible cell codes, however, is likely greater than that seen in evolved
organisms because the historical process of evolution is not expected to allow
exploration of every cell code (i.e., $C_{tot}>Cell\hskip
3.0ptCode_{tot}>Cell\hskip 3.0ptCode_{evol}$).

Cell codes of varying complexity have evolved over time to specify the
development of each organism that has ever existed [6]. Cell codes could in
principle differ in the relative amounts of information stored in the genome
sequence versus in the arrangement of molecules. The interdependence of these
two stores of information invite exploration of the relationship between their
storage capacities over evolutionary time. Consider the consequences of adding
into a pre-existing cell code a newly evolved gene sequence that codes for a
protein. The number of possible genome sequences of a given length that can
support this cell code decreases because fewer distinct genomes can include
the gene sequence for the new protein (i.e., total sequences becomes less than
$X^{L}$, Supplemental Figure 2). However, the number of distinguishable
arrangements of all molecules can either increase or decrease. Increase can
occur because addition of the new DNA sequence, the transcribed RNA, and the
translated protein to the contents of the cell could all lead to new
interactions with pre-existing molecules (i.e. $E$, $S$, and $P$ could all
increase, resulting in a larger value for
$\sum_{i}E_{i}\sum_{j}S_{j}\sum_{k}P_{k}$). Decrease can occur because these
new molecules could constrain the arrangement through regulatory interactions
(see section titled Entity-Sensor-Property: insights). Furthermore, the
magnitude of changes in $\sum_{i}E_{i}\sum_{j}S_{j}\sum_{k}P_{k}$ depends on
the nature of the new gene product (e.g., expression or repression of many
gene sequences by a transcriptional activator or repressor, respectively,
could lead to large changes). Studies on the origin and evolution of
information storage could illuminate trends in the partition of heritable
information between different molecular stores and lead to general principles
(for related views emphasizing arrangement see [21, 22], genome sequence see
[23, 24], anatomy see [25], and energy see [26]). For example, the complexity
of cell codes, and thus organisms, may have increased through restriction of
the genome sequence along with expansion of the arrangement of molecules as
sources of neutral or adaptive variation.

Entity-Sensor-Property: extensions

Several processes in living systems could limit or expand the number of
arrangements in the bottleneck stage
($\sum_{i}E_{i}\sum_{j}S_{j}\sum_{k}P_{k}$). Processes that can change the
information content of cell codes by decreasing (e.g., self-organization and
self-assembly), increasing (e.g., chemical modification) or variably changing
(e.g., compartmentalization) entities, sensors, and/or properties are being
actively analyzed. Living systems could manipulate heritable information
through the regulation of all such processes.

Impact of self-assembly and self-organization. Order can arise through the
spontaneous association of molecules in living systems. Two forms of such
spontaneous order have been recognized: (1) self-assembly, which refers to the
formation of static structures that are relatively stable (e.g., viruses,
flagella)[27]; and (2) self-organization, which refers to the formation of
dynamic structures that appear stable (e.g., cytoskeleton, endocytic
compartments) [28]. Both forms of order, however, depend on the immediate
molecular environment. Therefore, changing the surroundings of a ‘self-
assembled’ or ‘self-organized’ structure can result in alternative
configurations that may be distinguishable by evolved sensors. For example,
cells can use an adaptor protein to modulate the size of vesicles that form
through self-assembly [29] and cells can respond to pressure by reversibly
disassembling the mitotic spindle that is maintained through self-organization
[30, 31]. In this way, living systems can store and retrieve information from
self-assembled and self-organized collections of molecules.

Impact of chemical modifications. Modifications of nucleic acids (5mC, 5hmC,
m6A, etc.) or proteins (phosphorylation, methylation, glycosylation, etc.)
result in new entities with properties that could potentially be measured by
sensors. Modified bases on the genome could increase the number of possible
spatial arrangements of the genome and its binding partners (i.e., $E$, $S$,
and $P$ in equation(1)), and could also increase sequence information (i.e.,
$X$ in equation (1)) if the modification alters base-pairing. Modifications on
RNA or proteins on the other hand could either increase or decrease $E$, $S$,
and $P$, but always reduce the maximal number of genomes of a given length
that could support such a modification because each possible genome would be
constrained to include the gene sequence for the enzyme that catalyzes the
modification (i.e. total sequences become less than $X^{L}$). Similar
considerations hold for modifications of all other molecules in the bottleneck
stage.

Impact of compartmentalization. Living systems dynamically manipulate which
entities come together into organized units and which outputs from these units
are subsequently measured. When different subcellular compartments form, the
same entity or sensor could be present in two or more different compartments.
If two such pools of the same entity are sensed separately during the life
cycle of an organism, the total number of possible configurations are
effectively increased. Alternatively, many different entities could be
encapsulated into one compartment. If only a few aggregate properties of the
compartment are sensed during the life cycle of an organism (e.g. droplet
sizes of phase-separated aggregates such as RNA granules [32] or numbers of
organelles such as mitochondria), the number of distinguishable configurations
are effectively reduced.

These different ways of changing entites, sensors, and properties highlight
the multiscale nature of living systems and suggest the utility of different
Entity-Sensor-Property frameworks at different scales and across scales.

Entity-Sensor-Property: insights

To appreciate some implications of the framework, consider a toy model where
the genome sequence and the environment are held constant (Figure 2).

Let the remaining contents of a ‘cell’ include three entities
($E_{1},E_{2},E_{3}$ \- three english letters) that can be at four different
states (two fonts with upper and lower cases) and be sensed by two sensors
($S_{1}$ measuring lines and $S_{2}$ measuring curves). Each state is
analogous to different experimentally measurable values for a property of
molecules in a cell (e.g., concentration, localization, shape, charge, etc.).
Consider the entity $E_{1}$ in state ‘A’ made of three straight lines. A
sensor that measures lines could measure one of numerous possible properties:
thickness of lines, color of lines, length of lines, etc. For simplicity, let
number be the only property $P$ sensed by both $S_{1}$ and $S_{2}$. For
example, the value of the property sensed by $S_{1}$ of $E_{1}$ in state ‘A’
is 3 and that sensed by $S_{2}$ of $E_{3}$ in state ‘c’ is 1 (see Figure 2 for
all values).

Sensors can limit information storage. To calculate the relevant information
stored in a system, we need to enumerate the number of different states of the
entities sensed by the system (Figure 2).

Figure 2: Figure 2. Distinguishable states in a toy model of possible cell
codes with a given ‘genome’ and ‘environment’. Three entities
($E_{1},E_{2},E_{3}$), two sensors ($S_{1},S_{2}$), and one sensed property
($P$) are considered. The measurable property values of each entity by each
sensor is enumerated ($E_{1}S_{1}P,E_{2}S_{2}P,...$). Each distinguishable set
of property values for all entities defines a distinguishable state.
Therefore, the number of distinct elements in a set of the measured values
(i.e $\left|E_{1}S_{1}P\right|,\left|E_{1}S_{2}P\right|,...$) can be used to
calculate the total number of distinguishable states in the system ($4\times
3\times 2\times 2\times 3\times 2=288$), which is less than the number
expected if every value of every entity were distinguishable ($4\times 4\times
4\times 4\times 4\times 4=4096$).

While each sensor can sense one property of each entity, different states of
an entity may not always be distinguishable by a sensor (e.g. $S_{1}$ will
measure the states ‘C’ and ‘c’ of $E_{3}$ as 0 and $S_{2}$ will measure both
as 1). Such indistinguishability is evident in living systems as the
requirement for threshold levels of a signal for a detectable response. Thus,
‘thresholding’ by the sensor results in a reduction in the total number of
states of the system and thus the storable information (Figure 2). For
example, this system of $E$, $S$, and $P$ can only distinguish 288 states
($\approx$8.17 bits), but a naive estimate based on the ability to distinguish
all states of all entities yields 4096 states (12 bits). Thus, the number of
entropic states of a system depends on the available sensors and their
sensitivity.

Distinct states may be equivalent. Selection can impose external constraints
on the form and function of a living system. For example, the environment of
the system in the toy model might require ‘cells’ with consistent fonts and
case for survival. This would result in the survival of cells with ‘ABC’,
‘abc’, etc. as the values for each of the three entities (Supplemental Figure
3). However because there are two possible cases (upper vs. lower), there
would be two distinguishable states that are effectively equivalent for
survival in this environment. Such situations could result in unregulated
redundancy such that similar functions are performed by different molecules in
random sets of cells [33]. Over evolutionary time scales, this type of
unregulated redundancy could result in organisms with similar form and
function but different underlying molecular mechanisms [34]. These
considerations also hold when such equivalency is imposed by sensors that fail
to distinguish different entities. For example, a channel protein responding
to changes in membrane potential would measure changes in different ions as
equivalent as long as the end result was a similar change in potential [35].

Figure 3: Figure 3. Regulation reduces the number of states that can be sensed
by the system. (A) Two states for each entity - high (upper case) or low
(lower case) - were considered for exploring the impact of regulation in the
toy model. (B) Consequences of introducing regulatory constraints. Inhibition
(bar) or activation (arrow) between all pairs of entities were considered.
Matrices of distinguishable values (product of all = number of states) for
each cell with regulatory interactions between $E_{1}$ and $E_{2}$ (top),
$E_{1}$ and $E_{3}$ (middle), or $E_{2}$ and $E_{3}$ (bottom) are shown.
Different regulatory constraints result in differential reduction in the
number of states of the system.

Regulation reduces sensed states. The different states of each entity could be
classified as high (upper case) or low (lower case) to simplify the analysis
of regulation in the system (Figure 3A). This simplification is similar to
Boolean networks that have been used to explore the impact of regulation [36].
All additions of either activation or inhibition as a regulatory interaction
between two entities reduces the number of distinguishable states in the
system (Figure 3B). This reduction occurs because any regulatory interaction
between two entities couples changes in those entities. As a result, two
entities that were previously free to vary independently become either
directly or inversely correlated, leading to an overall reduction in the
number of possible states. Different regulatory architectures can lead to
different states with equivalent capacity for information storage.
Specifically, 12 different single regulatory interactions in the toy model
lead to only 3 different storage capacities - 96, 216, or 256 states (Figure
3B). Adding two regulatory interactions results in all 36 different regulatory
architectures having only 96 distinguishable states (Supplemental Figure 4).
These results suggest a preliminary conclusion: regulation reduces the ability
of systems to store information in the arrangement of molecules.

Reducing states may promote robustness.  Robustness is the ability of living
systems to remain similar despite some variation introduced by environmental
or internal conditions [37]. In other words, some changes either do not alter
anything about a robust system or can alter some entities but nevertheless do
not substantially affect the system. The differences in the number of states
in cells with different regulatory architectures (Figure 3B) suggest a
relationship between regulation and robustness of cell types. Unlike in the
toy model, in living systems, all sensors are made from entities (equation
(1)). Therefore, cell types with fewer states could be more robust because
they are only capable of sensing, and thus responding to, fewer perturbations.
Conversely, cell types with many states could be less robust because they are
capable of sensing and responding to many perturbations. Changes in regulatory
architectures could therefore be used to generate cell types that are
differently responsive to external signals, which may have implications for
the observed robustness of development [38]. To achieve such robust
development, entities need to be assembled into cell codes such that the same
sequence of events unfolds despite some perturbations. Storing entities as
perturbation-resistant assemblies or combining entities that fail under some
conditions with entities that fail under other conditions (redundancy) are
possible ways to ensure robust cell codes and subsequent development. An
additional possibility suggested by these observations is reducing the number
of sensors through increased regulation such that some perturbations are
simply not sensed.

Two-base genomes could be part of efficient living systems

Our current ability to exquisitely edit genomes and transcriptomes [39, 40] is
a limited manipulation of living systems in that the outcome of the edit is
entirely determined by how the living system interprets the change. In other
words, we can make changes to a sequence and read out what the living system
does with the changed sequence but we cannot yet make changes that instruct a
living system to perform arbitrary tasks. Such expanded manipulation could
require ways of increasing the complexity of the stored heritable information.
As suggested by equation (1), this increase could be achieved by either
increasing storage in the genome sequence or by increasing storage in the
arrangement of molecules. Increases in the storage capacity of a genome by
increasing the number of different bases will require concomitent increases in
the complexity of the machinery for accurate reading and writing of the
genome. For example, a 16-base genome of length $L$ has four times the
capacity of a 2-base genome ($4L$ bits vs $L$ bits).

Figure 4: Figure 4. Increases in the types of base pairs in a genome increase
the maximal information that can be stored per base but decrease the mean
variation in base availability that can support the high information storage.
Plot showing how maximal information ($H$) of a base pair varies with the
standard deviation ($\sigma$) of base probability. For each n-base system (n =
2, 4, 6, 8, 10), the results for one million base probabilities drawn form a
uniform distribution are plotted.

However, such a genome would require machinery for discerning eight times as
many kinds of bases. Furthermore, the range of availabilities of bases that
can support the enhanced information storage decreases with an increase in the
number of different bases (Figure 4).

Perhaps the simplest route to the synthesis of living systems with arbitrary
capacity to store heritable information would be to use a longer 2-base genome
that is equivalent to the natural 4-base genome (e.g. needs 5-base codons for
encoding at least 20 amino acids, $2^{5}>20>2^{4}$), but can be supported by
simpler machinery in the cell to read and write the genome. Testing the
practicability of this speculation requires systematically changing the
chemistry of the genome and the cell while preserving overall storage
capacity.

Discussion

By jointly considering all information transmitted from one generation to the
next using molecules, I have developed an expanded view of heredity (see
supplemental text for other applications). Heritable information stored
outside the genome sequence is limited by mutual constraints with the
sequence, by regulatory architectures, and by what a living system can sense
about itself and its environment.

Strengths and limitations of framework. Entities and sensors in a cell were
parsed based on their roles at a particular time in development - the
bottleneck stage. However, the roles of entities and sensors are potentially
interconvertible over time. A sensor could become a unresponsive entity for a
while and an entity could become a responsive sensor when it encounters
another appropriate entity. Such changes in roles are likely part of the
changes during development that lead to the assembly of cell codes at the
start of each generation. Given this time-bound nature of entities and
sensors, what is the duration of a bottleneck stage? This question is
currently very difficult to answer and poses a practical problem for
unambiguously defining the cell code of an organism. Nevertheless, the
stability of cell types suggests that functionally important states are
preserved for significant periods through homeostasis.

The framework presented here does not account for the stochastic and noisy
nature of all interactions within a cell. For example, there are fundamental
limits to control that result from information loss [41] and the physical
limits of biochemical signaling [42, 43]. Unlike in man-made communication
systems, the presence of numerous simultaneous signaling pathways in living
systems - including as yet unknown pathways - makes it unclear whether any
observed variation in one signaling pathway should be characterized a priori
as interference from another signaling pathway or as noise. Nevertheless,
developing an understanding of heredity in terms of genome sequence, entities,
sensors and properties is a first step towards future extensions of the
framework that could address these issues.

Some past frameworks for analyzing living systems provide conceptual
structures for explaining their evolution and behavior but do not inform their
construction or origin. Models that analyze evolutionary outcomes regardless
of the material basis of genotype and phenotype (e.g., ref. [44]) are useful
guides for the analysis of organisms at the population level but not for the
construction of organisms from molecules sought here. Phylogeny, architecture,
and adaptation have been combined to understand trends in the evolution of
form [45], but such models are currently not fine-grained enough to enable
construction. The productive analysis of complex systems by partitioning a
system into abstract nodes and edges to view particular aspects of living
systems as networks [46] has generated intuitions and approaches that could be
extended to the framework presented here. Such extension beyond abstract
networks is necessary to enable the construction of living systems because
typical abstractions do not incorporate all relevant properties of cellular
contents. The explicit consideration of relevant properties for all entities
that are measured by sensors in the framework presented here could help in
accruing knowledge in a form that is useful for the construction of living
systems and for the realization of a practical systems biology [47, 48].

Synthesis of living systems. Building something using its constituent parts is
a good way to discover the flaws in our understanding of how it is put
together. For example, it is currently unclear if perfect self-replication
ever occurs in living systems. Perhaps the perpetuation of life is always
associated with having entities that are not recreated with a period of one
generation but rather with longer or shorter periods. For example, when the
noisy and variable behavior of a synthetic oscillating circuit in E. coli [49]
was improved to obtain synchronous long-term oscillations [50], the period of
oscillation increased to 14 generations. Such possibilities can be explored by
allowing different generation times for the precise recreation of some
entities and arrangements in the cell code. The similarity in form and
function of parent and progeny, however, suggests that the cell codes
recreated with a period of one generation are at least nearly equivalent for
specifying development in each generation.

Evolved cell codes are unlikely to be efficient stores of heritable
information because of the historical measures and counter-measures through
which evolution proceeds [51, 52]. Efficient storage of the mutual information
between two variables can be achieved using a compressed bottleneck variable
[53, 54]. If there was selection for effectively packing maximal information
into the bottleneck stage in living systems, the entities and arrangements of
evolved cell codes could similarly be efficient stores of the mutual
information between the past and the future. All such efficient cell codes
might have similar characterstics as observed in cellular automata in which
the capacity to support computation emerges (captured in the $\lambda$
parameter in [55]). Despite the possibility of such overall optimization, it
is unclear if living systems can evolve to maximally optimize information
storage and/or transmission for a particular trait. In fact, it might be
difficult to define what the ‘optimum’ is for a process because the presence
of many homeostatic mechanisms in cells, including transgenerational
homeostasis [6], require opposing processes that could limit optimality.
Experimental approaches that attempt to generate minimal bacterial cells [56]
need to be extended to different organisms to discover how the complexity of
organisms scales with their cell codes.

Making efficient living systems of arbitrary complexity requires a holistic
approach to information handling. The joint cosideration of all heritable
information presented in this article suggests that a genome with two
different kinds of bases might function as an efficient replicating store when
combined with the simplest possible cycling stores (Figure 4). Thus far,
experimental approaches to fundamentally change heritable information have
focused on increasing the storage capacity of the genome. A 50% increase in
the storage capacity of DNA sequence can be achieved by doubling the number of
different bases in DNA [57]. Furthermore, an organism that uses a 4-base
genome can be modified with two additional DNA bases to successfully store
[58] and retrieve [59] information. In contrast, we cannot yet engineer such
increases in the information stored by the arrangement of molecules because
our knowledge of this store of heritable information is in its infancy. The
theoretical and practical limits of varying all heritable information deserve
exploration to understand the evolution of natural, modified, and synthetic
living systems.

Acknowledgements

I thank Tom Kocher, Karen Carleton, Charles Delwiche, David Wolpert, Chris
Kempes, Michael Lachmann, Artemy Kolchinsky, James Yorke, Daniel Damineli,
Pierre-Emanuel Jabin, Katerina Ragkousi, David Jordan, KP Mohanan, LS
Sashidhara, Sudha Rajamani, Jyotsna Dhavan, Michael Levin, Alejandro Sánchez
Alvarado, Ajay Chitnis, and Victor Ambros for discussions and encouragement;
and Tom Kocher, Karen Carleton, Ken Helland, members of the Jose lab, and an
anonymous reviewer for comments on the manuscript. Research in my lab is
supported by the NIH (R01GM111457 and R01GM124356).

References

## References

  * [1] Gasper Tkačik and William Bialek  “Information Processing in Living Systems”  In _Annual Review of Condensed Matter Physics_ 7, 2016, pp. 89–117
  * [2] Masahiro Tanaka  “A physical characterization of biological information and communication system model of ecosystems”  In _Journal of Theoretical Biology_ 110, 1984, pp. 89–117
  * [3] Paul Nurse  “Life, logic and information.”  In _Nature_ 454, 2008, pp. 424–426
  * [4] Antoine Danchin  “Information of the chassis and information of the program in synthetic cells”  In _Systems and Synthetic Biology_ 3.1-4, 2009, pp. 125–134
  * [5] Sydney Brenner  “Sequences and consequences”  In _Philosophical Transactions of the Royal Society B_ 365, 2010, pp. 207–212
  * [6] Antony M Jose  “Replicating and cycling stores of information perpetuate life”  In _BioEssays_ 40.4, 2018, pp. 1700161
  * [7] Gérard Battail  “Heredity as an Encoded Communication Process”  In _IEEE Transactions on Information Theory_ 56.2, 2010, pp. 678–687
  * [8] Claude Shannon  “A mathematical theory of communication”  In _The Bell System Technical Journal_ 27.3, 1948, pp. 379–423623–656
  * [9] Thomas M Cover and Joy A Thomas  “Elements of information theory”  Hoboken, New Jersey: Wiley-Interscience, 2016
  * [10] Moshe Sipper  “Fifty years of research on self-replication: an overview”  In _Artificial Life_ 4, 1998, pp. 237–257
  * [11] Umberto Pesavento  “An implementation of von Neumann’s self-reproducing machine”  In _Artificial Life_ 2.4, 1995, pp. 337–354
  * [12] Christopher G Langton  “Self-reproduction in cellular automata”  In _Physica D_ 10.1-2, 1984, pp. 135–144  DOI: http://golly.sourceforge.net
  * [13] Michael J Rust, Susan S Golden and Erin K O’Shea  “Light-driven changes in energy metabolism directly entrain the cyanobacterial circadian oscillator”  In _Science_ 331, 2011, pp. 220–223
  * [14] Tetsuya Mori, Brian Binder and Carl Hirschie Johnson  “Circadian gating of cell division in cyanobacteria growing with average doubling times of less than 24 hours”  In _Proceedings of the National Academy of Sciences, USA_ 93, 1996, pp. 10183–10188
  * [15] Robert Y L Tsai and Ronald D G McKay  “A multistep, GTP-driven mechanism controlling the dynamic cycling of nucleostemin”  In _Journal of Cell Biology_ 168.2, 2005, pp. 179–184
  * [16] Katerina Ragkousi et al.  “Cell-cycle-coupled oscillations in apical polarity and intercellular contact maintain order in embryonic epithelia”  In _Current Biology_ 27.9, 2017, pp. 1381–1386
  * [17] Michael H Hastings, Elizabeth S Maywood and Marco Brancaccio  “Generation of circadian rhythms in the suprachiasmatic nucleus”  In _Nature Reviews Neuroscience_ 19, 2018, pp. 453–469
  * [18] Gerald F Joyce  “Bit by Bit: The Darwinian Basis of Life”  In _PLoS Biology_ 10.5, 2012, pp. e1001323
  * [19] Harold J Morowitz  “Some order-disorder considerations in living systems.”  In _Bulletin of Mathematical Biophysics_ 17, 1955, pp. 81–86
  * [20] Shinya Yamanaka  “Induced pluripotent stem cells: past, present, and future”  In _Cell Stem Cell_ 10.6, 2012, pp. 678–684
  * [21] Sara Imari Walker and Paul C W Davies  “The algorithmic origins of life.”  In _Journal of the Royal Society Interface_ 10, 2013, pp. 20120869
  * [22] Chris Fields and Michael Levin  “Multiscale memory and bioelectric error correction in the cytoplasmcytoskeletonmembrane system.”  In _Wiley Interdisciplinary Reviews: Systems Biology and Medicine_ , 2018, pp. c1410
  * [23] Michael Lynch  “The frailty of the adaptive hypothesis for the origins of organismal complexity.”  In _Proceedings of the National Academy of Sciences, USA_ 104.1, 2007, pp. 8597–8604
  * [24] Eörs Száthmary and John Maynard Smith  “The major evolutionary transitions.”  In _Nature_ 374, 1995, pp. 227–232
  * [25] Michael Levin  “The Biophysics of Regenerative Repair Suggests New Perspectives on Biological Causation”  In _BioEssays_ 42, 2020, pp. 1900146
  * [26] Arto Annila and Keith Baverstock  “Genes without prominence: a reappraisal of the foundations of biology.”  In _Journal of the Royal Society Interface_ 11, 2014, pp. 20131017
  * [27] D J Kushner  “Self-assembly of biological structures”  In _Bacteriological reviews_ , 1969, pp. 302–345
  * [28] G Nicolis and I Prigogine  “Self-organization in nonequilibrium system.”  New York, New York, U.S.A.: John Wiley & Sons, Inc., 1977
  * [29] Bing Zhang et al.  “Synaptic vesicle size and number are regulated by a clathrin adaptor protein required for endocytosis.”  In _Neuron_ 21.6, 1988, pp. 1465–1475
  * [30] Edward D Salmon  “Pressure-induced depolymerization of spindle microtubules. I. Changes in birefringence and spindle length.”  In _Journal of Cell Biology_ 65, 1975, pp. 603–614
  * [31] Edward D Salmon  “Pressure-induced depolymerization of spindle microtubules. II. Thermodynamics of in vivo spindle assembly.”  In _Journal of Cell Biology_ 66, 1975, pp. 114–127
  * [32] Yongdae Shin and Clifford P Brangwynne  “Liquid phase condensation in cell physiology and disease”  In _Science_ 357.6357, 2017, pp. eaaf4382
  * [33] Shusha Ravikumar, Sindhuja Devanapally and Antony Jose  “Gene silencing by double-stranded RNA from C. elegans neurons reveals functional mosaicism of RNA interference.”  In _Nucleic Acids Research_ 47.19, 2019, pp. 10059–10071
  * [34] John R True and Eric S Haag  “Developmental system drift and flexibility in evolutionary trajectories”  In _Evolution and Development_ 3.2, 2001, pp. 109–119
  * [35] Michael Levin  “Molecular bioelectricity in developmental biology: new tools and recent discoveries”  In _BioEssays_ 34.3, 2012, pp. 205–217
  * [36] Stuart A Kauffman  “Metabolic stability and epigenesis in randomly constructed genetic nets”  In _Journal of Theoretical Biology_ 22, 1969, pp. 437–467
  * [37] Jorg Stelling et al.  “Robustness of cellular functions”  In _Cell_ 118, 2004, pp. 675–685
  * [38] Conrad H Waddington  “Canalization of development and the inheritance of acquired characters”  In _Nature_ 150, 1942, pp. 563–565
  * [39] Gavin J Knott and Jennifer A Doudna  “CRISPR-Cas guides the future of genetic engineering”  In _Science_ 361.6405, 2018, pp. 866–869
  * [40] Holly A Rees and David R Liu  “Base editing: precision chemistry on the genome and transcriptome of living cells”  In _Nature Reviews Genetics_ 19, 2018, pp. 770–788
  * [41] Ioannis Lestas, Glenn Vinnicombe and Johan Paulsson  “Fundamental limits on the suppression of molecular fluctuations”  In _Nature_ 467, 2010, pp. 174–178
  * [42] Howard C Berg and Edward M Purcell  “Physics of Chemoreception”  In _Biophysical Journal_ 20, 1977, pp. 193–219
  * [43] William Bialek and Sima Setayeshgar  “Physical limits to biochemical signaling”  In _Proceedings of the National Academy of Sciences, USA_ 102.29, 2005, pp. 10040–10045
  * [44] Olivier Rivoire and Stanislas Leibler  “A model for the generation and transmission of variations in evolution”  In _Proceedings of the National Academy of Sciences, USA_ 111.19, 2014, pp. E1940–E1949
  * [45] Derek E G Briggs  “Evolving form and function: fossils and development. Proceedings of a symposium honoring Adolf Seilacher for his contributions to paleontology, in celebration of his 80th birthday”  New Haven, Connecticut, U.S.A.: A Special Publication of the Peabody Museum of Natural History, Yale University, 2005
  * [46] Albert-László Barabási and Zoltán N Oltvai  “Network biology: understanding the cell’s functional organization”  In _Nature Reviews Genetics_ 5, 2004, pp. 101–113
  * [47] Hiroaki Kitano  “Systems Biology: a brief overview.”  In _Science_ 295.5560, 2002, pp. 1662–1664
  * [48] Hiroaki Kitano  “Computational Systems Biology.”  In _Nature_ 420, 2002, pp. 206–210
  * [49] Michael B Elowitz and Stanislas Leibler  “A synthetic oscillatory network of transcriptional regulators”  In _Nature_ 403, 2000, pp. 335–338
  * [50] Laurent Potvin-Trottier, Nathan D Lord, Glenn Vinnicombe and Johan Paulsson  “Synchronous long-term oscillations in a synthetic gene circuit”  In _Nature_ 538, 2016, pp. 514–517
  * [51] Francois Jacob  “Evolution and Tinkering”  In _Science_ 196.4295, 2018, pp. 770–788
  * [52] S J Gould and R C Lewontin  “The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptationist programme”  In _Proceedings of the Royal Society B_ 205, 1979, pp. 581–598
  * [53] Naftali Tishby, Fernando C Pereira and William Bialek  “The information bottleneck method”  In _arXiv_ , 2000  DOI: arXiv:physics/0004057 [physics.data-an]
  * [54] Artemy Kolchinsky and David H Wolpert  “Semantic information, autonomous agency and non-equilibrium statistical physics”  In _Interface Focus_ 8, 2018, pp. 20180041  DOI: 10.1098/rsfs.2018.0041
  * [55] Chris G Langton  “Computation at the edge of chaos: phase transitions and emergent computation.”  In _Physica D_ 42, 1990, pp. 12–37
  * [56] Clyde A Hutchison III et al.  “Design and synthesis of a minimal bacterial genome”  In _Science_ 351.6280, 2016, pp. aad6253
  * [57] Shuichi Hoshika et al.  “Hachimoji DNA and RNA: A genetic system with eight building blocks”  In _Science_ 363, 2019, pp. 884–887
  * [58] Yorke Zhang et al.  “A semisynthetic organism engineered for the stable expansion of the genetic alphabet”  In _Proceedings of the National Academy of Sciences, USA_ 114.6, 2017, pp. 1317–1322
  * [59] Yorke Zhang et al.  “A semi-synthetic organism that stores and retrieves increased genetic information”  In _Nature_ 551, 2017, pp. 644–647