# Considering Hilbert’s Approach: Problems from a Cell’s Perspective

Nima Dehghani  MIT, Cambridge, MA, USA<EMAIL_ADDRESS>
###### Abstract

Examining individual components of cellular systems has been successful in
uncovering molecular reactions and interactions. However, attempts to combine
these discrete components into a system-scale map have failed due to missing
links (unknown variables), contemned nonlinearities in high-dimensional
parameter space, downplayed natural noisiness and stochasticity, ignored
causal influence and disregarded the temporal dynamics. Composite static and
phenomenological descriptions, while may look complicated, lack the essence of
what makes the biological systems truly “complex”. Formalization of system-
level problems is key to constructing a meta-theory of biology. This work aims
to identify a few problems that cells must solve as a template for such
formalization and as a step towards axiomatizing biological investigations.

Theory, Model, Fromalism, Phenomenology, Axiomatization

> Keywords: Theory, Model, Fromalism, Phenomenology, Axiomatization

Scientific journals and textbooks often contain graphical representations of
cellular mechanisms. These can range from 3D cartoons of subcellular
structures and their interactions to diagrams of signaling pathways. Advances
in imaging, high throughput gene expression methods, and our better
understanding of biochemical pathways have led to an abundance of images
showing extremely complicated yet orderly intracellular structures and mind-
bugling yet engineered-like renditions of signaling and communication.
However, it is important to recognize that these types of images and graphs
are only surmising abstractions of what is really happening in cellular
systems. Each step in a pathway may represent the work of a PhD dissertation
or countless hours of research on a small set of molecules.

It is difficult to capture the complexity of cellular interactions with static
descriptions. Given the high degree of complexity and stochasticity of the
intracellular events, it is remarkable that cells and simple organisms are
able to achieve such high levels of reliability and robustness. How do cells
control their inner dynamics and process information about their environment?
How do they respond to challenging conditions? Or if one wants to give these
questions an anthropomorphic flavor, how does a cell know what to do in
challenging environments? Can we understand the mechanisms or goals behind
cellular behavior through reductionist or ateleological approaches? Or as
Lazebnik elegantly titled his essay "can a biologist fix a radio" Lazebnik
(2002)?, can we truly understand the inner workings of a cell?

One approach to understanding cellular behavior is to adapt methods from
fields such as physics or computer science. However, rather than simply
applying existing methods, it may be more effective to take an axiomatic
approach. This would involve framing the questions correctly and then
searching for or developing new methods to understand “evolved” biological
systems. Combining methods from various fields such as dynamical systems
theory, computer science, statistical inference and machine learning may also
be helpful Vit (2022). Instead of focusing on small problems with available
methods, it may be more productive to define a range of problems that a cell
must solve and develop a systematic theoretical approach. This could help
shape our thinking in a well-defined paradigm, similar to how Hilbert’s list
of unsolved problems in mathematics influenced the field Hilbert (1902). As a
result, instead of being limited by the methods in which we are experts, our
thinking can be guided by a well-defined paradigm.

In an attempt to formalize some key problems, here, we provide an expose of
three fundamental challenging issues that cells face: finding the right
control switches, solving the need for reconfiguration, and dealing with
internal noise. Previous attempts at axiomatizing biology, such as those by
Woodger Woodger (1962) as well as Mahner and Bunge’s metaphysically-schemed
approach to biology Mahner and Bunge (1997) are prime examples of failed
attempts at axiomatization of biology. This failure is due to shortcomings in
proper formalization of the key problems in biology. To avoid repeating these
mistakes, it is important to properly formalize key issues before attempting
to develop an axiomatic approach. By working within a paradigm and adapting it
as we progress (both experimentally and theoretically), we can build a bridge
from biological intuition to mathematical developments to further our
understanding of living systems.

## Problem 1: Where are the control switches?

Single cell organisms must constantly adapt to changing environments, such as
fluctuations in temperature, pH, and the concentration of necessary substrates
or harmful chemicals. Multicellular systems have more internal stability due
to sophisticated regulatory systems, but their individual cells must also
overcome fluctuations in their constrained environments. How do cells, either
in their unicellular or multicellular form, maintain their internal stability?
To maintain internal stability, cells must find the relevant switches to adapt
to changes in their environment. Effective regulation must satisfy two
constraints: the cell must find the relevant switches without causing
permanent damage or death, and it must do so in a timely manner due to limited
time and energy. Therefore, adaptive cellular regulation is a time-constrained
optimization in search of the relevant switches for the necessary actions.

Finding the right switch at the right time is a complex problem due to the
large number of switches in a cell. One possible solution is to use a trial
and error approach, as described by Ashby Ashby (1945, 2013). He proposed a
problem in which a person must find the right configuration of 1000 switches
to turn on a light. How long does it take to find the right configuration?
Three possible scenarios have vastly different execution times. A simple
through search will take $2^{1000}$ seconds (2 seconds for each key’s on and
off choices) and therefore is impractical. A serial test with knowledge of
partial correctness would take around 500 seconds on average. Testing all
switches individually but in parallel would take only 1 second. Karl Popper
considered “trial and error” to be the source of all knowledge under
evolutionary epistemology Popper (2005). While “trial and error” may work for
high-dimensional problems such as adaptive cellular regulation, there is a
catch: if a cell attempts trial and error, it may turn on a wrong switch, turn
off a needed one, or run out of time to turn on the right switch at the right
time. Resorting to trial and error is likely to be fatal for the cell.

One could argue that in a large population of cells, there may be one that
successfully adapts through trial and error. However, the structure of
switches within a cell is not random. There is a hierarchy of switches and
regulatory motifs Yu and Gerstein (2006); Tonner  _et al._ (2015), with some
operating at lower scales and others modulating them at higher scales. This
translates to a causal structure where the right configuration can be achieved
more efficiently through natural selection or within the lifetime of an
organism. Either way, a straightforward “trial and error” is not efficient for
dealing with changes in a constrained fashion. The combinatorial complexity of
the switch configuration problem means that a trial and error approach is not
only computationally unaffordable, but also made more difficult by a lack of
input information Bei, Chen, and Zhang (2013). Therefore to reach a desirable
problem-specific non-optimal solution that would require no/little knowledge
of the sources of external changes, the switch configuration will be best
handled if a cell itself can separate levels of control at macro vs micro
scales. Interestingly, phenomenological models can describe a system at the
macros scale but don’t tell us what exactly is happening at the micro level.
We can deduce that in order to understand how cells find the right switches at
the right time, it is necessary to have simultaneous measurements at both
macro and micro scales. These scales can not be examined separately and then
stitched together in a fixed diagram. Overlapping complementary measurements
at multiple scales can help construct a map of the causal structure of
switches and how the cell exerts internal control, rather than creating an
ever-growing chart of imaginary blueprints.

The multiscale complexity of protein and genetic switches presents challenges
for understanding the causal structure of control switches. In the case of
gene regulatory networks, this task is made more difficult by limitations in
the type of data needed to unravel the network’s causal structure Ahmed, Roy,
and Kalita (2020). Current efforts in methodological innovation aim to address
the issue of sparse and noisy gene expression time-series Aalto  _et al._
(2020); Lu  _et al._ (2021), but limited data has also led to the development
of inefficient and unreliable inference techniques Ahmed, Roy, and Kalita
(2020). As more high quality gene expression time-series become available, it
becomes imperative to focus on the development of causal inference methods
that match the multiscale nature of the (protein/gene) switch networks. For
example, in transcription multiregulatory systems, low-demand genes are mostly
controlled by repressors while high-demand genes have activator controllers
Shinar  _et al._ (2006). This error-load minimization mechanism may have been
the evolutionary driver of the emergence of various regulatory motifs Shinar
_et al._ (2006); Alon (2007). A search method for causal structure may need to
consider searching for these motifs. Dependence of the (regulatory)
information transfer on weak links, as opposed to the predominance of strong
links utilized in energy transfer and metabolic pathways, may have played a
significant evolutionary role in generating new activating or inhibiting
signals Kirschner and Gerhart (1998). The topological features of the
regulatory modules point to the existence of different connectivity structures
depending on the level of exposure to environmental variability and noise.
Lower connectivity limits the functional coupling of the network components,
promotes robustness, and is more abundant in the modules that are tightly
linked to the environmental variability. In contrast, modules that are
isolated from the external noise sources do not have the strong evolutionary
drive for sparse connections Navlakha  _et al._ (2014).

Biological systems have evolved features to minimize errors and remain robust
to environmental and signaling noise. This suggests that we need methods to
capture causality from a functional hierarchical perspective. Simply knowing
the connection between the components is not enough a causal map. Instead, we
will need to map the experimental results to a type of causal probing and in-
circuit testing that can capture different scales of interaction. Pearl’s
theory of causal and counterfactual inference based on structural models Pearl
(2000) and its recent extension to information theory Hoel, Albantakis, and
Tononi (2013); Hoel and Levin (2020) offer tools for quantitative analysis of
causal structure that can shed light on macro-scale models of network
interactions. However, time-delays and partial ordering of the processes that
are the result of genetic switch configurations, pose a serious challenge in
deciphering the causal structure of intracellular switches. As they stand,
current mixed formalisms of information theory and causal modeling are not fit
to handle the unbearable complexity of time-delays in a vast network of
interacting genes and proteins. Concurrency was developed to solve the problem
of partially ordered components in distributed systems synchronization Lamport
(1978). Interestingly, some concurrency-related methods such as $\Pi$-Calculus
and Petri nets were adapted to model cellular signaling pathways Regev,
Silverman, and Shapiro (2001); Priami  _et al._ (2001); Pinney, Westhead, and
McConkey (2003); Phillips and Cardelli (2007); Priami (2009). Fusing together
partially-ordered operations, causal modeling, and information theory may help
match the nature of biological switches and their interwoven multiscale
dynamics. The key notion is not to approach this problem as “what could
experts in method X can do for biology?”, but rather to first scope the
biological problem and cellular/subcellular constraints before constructing
relevant and robust methods.

## Problem 2: How to manage the need to reconfiguration?

We previously discussed how cells maintain their internal state by managing
gene/protein switch configurations. With minimal changes in the extracellular
environment, a homeostatic response can handle internal regulation through a
few adjustments. Homeostasis, initially conceptualized by Claude Bernard and
later formally defined by Walter Cannon Cooper (2008); Cannon (1929), deals
with a steady-state goal as a core regulatory mechanism in living systems
Modell  _et al._ (2015). This mechanism often relies on negative feedback
Schneck (1987); Modell  _et al._ (2015). The tight relation between the
negative feedback and control was the cornerstone of the cybernetics in
defining how a living system can automatically oppose the changes forced upon
it Wiener (1965). In fact, cybernetics stretched this notion to define
purposive behavior, giving the feedback regulatory mechanism a teleologic
flavor Rosenblueth, Wiener, and Bigelow (1943). However, issues such as lag
time of response and dampening oscillatory offshoot led to conceptualizing
other control mechanisms such as integrative error minimization through
negative feedback Yi  _et al._ (2000). Although elegant mathematically, it was
unclear how integral feedback control could be realized biologically. This
issue was recently resolved through proposing a controller topology that could
reach robust adaptation in noisy intracellular networks Briat, Gupta, and
Khammash (2016); Aoki  _et al._ (2019).

Homeostatic regulation, whether through negative feedback or antithetic
integral feedback control, is key in maintaining internal regulation in
response to extracellular changes. However, some extracellular changes are
more extreme and need a different solution. A lesser known sister of
homeostasis, referred to either as _Allostasis_ McEwen (1998), _Heterostasis_
Selye (1973), or _Adaptive Homeostasis_ Davies (2016), exerts a transient
expansion or contraction of the homeostatic response bounds. Adaptive
regulation has two important aspects. First, the need for response to extreme
events (for example a sudden but long-term change in extracellular Na) not
only pushes the cell to find the right switch configuration to maintain
internal stability, but it also mandates the need for long-term or lasting
reconfiguration (of gene and protein pathways). This prompts us to envision
the cell response as a multidimensional landscape with regulatory stable
states as basins of attraction. Exposure to extreme events can push the cell
to search for a new optimum basin in a new functional space with altered
regulatory bounds. Note that a newly reconfigured cell is no longer in the
same functional space as it used to be. The new functional space gives the
cell novel and altered regulatory bounds that change the intracellular
response time and/or nature of the response. Reconfiguration can be short or
long-term depending on the duration and severity of the external changes. More
complex cells may have a higher functional repertoire, with more basins in the
functional space to visit and perhaps with a memory to revisit the optimal
basin quickly in re-exposure to the same extreme event. The second feature of
the allostatic regulation is its predictive nature helping to reduce the
magnitude of the response error and needed compensation Sterling (2012).
Together, homeostasis and allostasis create a functional space where
phenotypic stability and plasticity characterize the dynamics and trajectory
of a cell’s configuration when exposed to varying degrees of extracellular
changes. Ion channels are a good example of how phenotypic stability and
plasticity work. These channels control the transport of ions across cell
membranes and are ubiquitous across unicellular and multicellular organisms.
Through this action, ion channels control the membrane potential and
intracellular ion concentration. Both of these properties, i.e. membrane
potential and intracellular concentration, are dependent on the number and
type of ion channels presently expressed in a cell. KscA, a well-known
bacterial potassium channel, is a tetramer with four-fold symmetry Doyle  _et
al._ (1998). It is a pH-gated potassium-specific channel, with a closed state
favored at neutral pH. The channel shows voltage-gated channel inactivation
and the conformational change of the energy landscape of gating is at its
minimum when the channel is closed Linder, de Groot, and Stary-Weinzinger
(2013). When exposed to an acidic environment, electrostatic changes in the
protein’s cytosolic domain cause the channel to open Hirano  _et al._ (2011);
Baker  _et al._ (2007). The length of time that the channel has been inactive
affects its reactivation, modulating its response to pH changes Gao  _et al._
(2005). Intracellular pH can affect many cellular mechanisms, including cell
metabolism, cell growth, Ca2+ homeostasis, proliferation and gene expression
Putnam (2012); Isfort  _et al._ (1993). Intracellular pH is regulated through
changes in membrane permeability to ions like K+, Na+, and H+ Booth (2007).
This complex regulation involves intricate patterns of nested and recursive
control. Extracellular acidity affects the cytosolic pH, changing the opening
probability and activation of potassium channels (such as KcsA) leading to a
departure from the energetically favored closed state putting an energy demand
on the cell. The duration of prior inactivation controls the speed at which
such change of functional space occurs. In parallel to voltage-gated K channel
response, other K transport systems play a key role in regulating the
cytosolic pH, and thus its effect on intracellular K concentration itself. For
example, if a cell has Kdp, a K transport system sensitive to both
extracellular K concentration and pH, its growth gets inhibited from a joint
reduction of Kdp activity and repression of Kdp gene expression due to
moderate external K+. In addition, at lower pH, the capacity of the cell to
regulate its cytosolic pH through K transport is undermined Roe  _et al._
(2000). its growth can be inhibited by a reduction in Kdp activity and
repression of Kdp gene expression. Here is where the theories from computer
science can help us in modeling and understanding these forms of
recursions.Likewise physics can guide us in untangling the dynamics at
multiple scales. Adapting and fusing these approaches may help us to better
understand the second problem (need to reconfiguration) that cells have to
deal with.

In more complex cells and multicellular systems, the intricate balance between
the phenotypic stability and plasticity has additional layers of complexity.
For example, steady-state distribution of ion channels is governed by a
correlated homeostatic regulation O’Leary  _et al._ (2013). Moreover, the
macroscopic electrogenic function can be maintained despite variation in ion
channel types Ori, Marder, and Marom (2018). On the other hand, more
complicated parameters set the functional landscape. For example, in neurons
there can be differential expression of Na and K channels between soma and
dendrites Duménieu  _et al._ (2017). Therefore the functional space and its
minima are uniquely defined for soma versus dendrites. This means the
functional space in dendrites or soma could be maintained within a pre-set
range while it might be altered for the other parts of the neuron. The result
of such differential plasticity is that the neuron as a unit of computation
can have varying overall function due to a polarized spatial balance between
phenotypic stability and plasticity. In addition, the intrinsic membrane
excitability can also be altered through a dynamic epigenetic modification of
DNA (through methylation) that controls the expression of ion channels and
synaptic response Meadows  _et al._ (2015, 2016). A similar situation is also
observed in the cardiac tissue where the balance between phenotypic stability
and plasticity is controlled by the variety of mechanisms at different levels
Rosati and McKinnon (2004). For example, repetitive pacing transforms the
electrophysiological attributes of cardiomyocytes to a new semi-stable state
Meadows  _et al._ (2015, 2016). This shift in the functional space takes place
through a reduction in the mRNA concentration of INa and ICa channels,
effectively altering their expression in cardiomyocytes Yue  _et al._ (1999).
The complex molecular mechanism for electrical remodeling of the cardiac cells
creates a major challenge for understanding how fine-tuning ion channel
expression controls cardiac excitability Balse  _et al._ (2012).Developing
mixed models of recursion and dynamics can help us better understand the
intricate causal nested chains of interaction that affect a cell’s functional
space post reconfiguration.

## Problem 3: How to harness noise rather than succumb to it?

Cells face the challenge of noise at almost every scale of operation. Noise
exists at the boundary with the external world as well as internally across
molecular interactions and physical interfaces. Despite the abundance of
noise, cells manage to function, grow and evolve. The viewpoint that noise is
pure nuisance has been gradually shifting to an understanding that noise is
not necessarily detrimental to biological systems Tsimring (2014). Although
discussions around noise could take volumes, here we will focus on certain
aspects of noise, how cells can handle such issues, and what modeling
framework can help us to better understand them. First, we will discuss the
boundary noise. The cell membrane is the boundary between the internal and
external world of the cell. Through molecular sensing and transport, cells
receive information about the external world and harvest needed chemicals.
Ligand-receptors and ion channels are the key molecular systems through which
a cell communicates and samples the external world. Whether a ligand is a
small molecule, an ion or a protein, the resultant conformational change in
the receptor protein starts the intracellular signaling that ensues
Bhattacharya  _et al._ (2008); Mary  _et al._ (2012). Aside from the occasions
when a ligand irreversibly binds to its receptor, the ligand-receptor
interaction is inherently stochastic and noisy Azpeitia, Balanzario, and
Wagner (2020). The irreversible binding may be desirable for drug testing
purposes Newman (1990), but in principle, is the opposite of what a continuous
information acquisition system would need. Considering the ligand-receptor
binding as the means for probing the extracellular space, one can think of the
“source (of the chemical), particles, and receptor” as an equivalent of
“transmitter, channel, receiver”. An information-theoretic treatment can guide
in separating different noise sources in a mixture model of kinetic and
stochastic reactions Pierobon and Akyildiz (2011).

An out of the box application of information-theoretic modeling of noise could
fail to capture many nuances that affect biological systems in various ways.
For example, the speed of ligand-receptor binding defines the nature of the
“noise” from a particular vantage point. When ligand and receptor binding and
unbinding happens very slowly, the response curve slowly reaches a
concentration-dependent saturation. As a result, the cell will be unable to
discriminate high levels of the ligands from each other. Therefore it is the
measurement of high concentrations of the ligand that are noisy from the
perspective of the cell. Surprisingly, this issue has led to the evolution of
cells that shift and expand their input dynamics to downstream events that
happen at a faster rate. This mechanism helps the receptor to reach
equilibrium binding, therefore bypassing the saturation Ventura  _et al._
(2014). Interestingly, if the number of receptors in a cell are low (which
introduces a different form of noise), this shift of response tuning to
downstream machinery prevents the receptor-level noise amplification Ventura
_et al._ (2014). Another special treatment of noise, at the boundary, occurs
when interference noise due to the presence of structurally-similar non-target
molecules adds up to the inherent stochasticity of a small number of target
molecules. In such situations, a ligand-induced stochastic cluster formation
of receptors takes place to enhance receptor’s sensitivity and specificity
Kajita, Aihara, and Kobayashi (2020). Another solution for noisy ligand-
receptor interactions is to synchronize the excitable receptors during signal
transduction Nagano (2020). These noise-induced spatial symmetry breaking or
temporal symmetry breaking might have separate evolutionary roots, but they
hint at a key point: cells are not passive receivers faulting in the abundance
of noise at the boundary. An important lesson is to recognize that the nature
of the noise itself can guide us to model and understand it effectively. As
Monod put it elegantly, biological systems are shaped by deterministic laws
and randomness Monod (1974). However, the way that living systems harness
noise (as exemplified above), shows us that a pure reductionist approach to
biophysical characteristics of ligand/receptor/membrane will leave us helpless
in understanding how cells manage challenging environments Noble (2021).

In the previous section, we talked about ion channels and regulations. Now,
we’ll delve deeper into the noise at the boundary of ion channels. The
development of patch clamp technology allowed for single-channel recording,
revealing that each channel’s conductance fluctuates as it switches between
open and closed states Hille (2001); Sakmann and Neher (1995). Thermal noise
is thought to be a likely source of such stochastic fluctuations Hille (2001);
Kispersky and White (2008).The probability of voltage-gated channels opening
and closing depends on the membrane potential, which affects channel noise.
This noise itself can disrupt the timing and generation of action potentials
and the ensuing membrane potential rise and reset White, Rubinstein, and Kay
(2000); Schneidman, Freedman, and Segev (1998). Channel noise can cause
significant variation of spiking propagation Horikawa (1991) and imposes a
lower limit on the miniaturization of cell signaling since the noise generated
by ion channels can shift the functional state of the cell (spontaneous action
potential generation) Faisal, White, and Laughlin (2005). While some aspects
of noise can be limiting or detrimental, noise itself can enhance the
detection of weak signals in certain nonlinear systems such as electronic
circuits and biological sensory systems Wiesenfeld and Moss (1995). A
phenomenon known as _stochastic resonance_ (SR) plays a critical role in
biological systems dealing with noise. SR was first introduced as an
explanation for the observed periodic occurrences of the Earth’s ice ages
Benzi, Sutera, and Vulpiani (1981). In SR, adding noise to a nonlinear
dynamical system can bring a weak signal above the threshold enabling the
system to detect sub-threshold signals Rouvas-Nicolis and Nicolis (2007);
Gammaitoni  _et al._ (1998). In that case, the system that is subject to
periodic forcing, shows a resonance in the spectrum that is absent in the
forcing and the perturbation Benzi, Sutera, and Vulpiani (1981). SR has been
shown to contribute to signal detection in various sensory systems, including
acoustic/electric stimulation of human hearing Zeng, Fu, and Morse (2000),
vision Simonotto  _et al._ (1997) and noise-enhanced tactile sensation and
balance control Priplata  _et al._ (2002); Collins, Imhoff, and Grigg (1996).
Large nonlinear networks are more sensitive to weak inputs when a fixed level
of noise is added. In such case, irrespective of the nature of the input
signal, it can be pushed to cross the threshold Collins, Chow, and Imhoff
(1995). However, SR is not limited to large networks of excitable units. SR
has also been observed at the cellular level in crayfish mechanoreceptor cells
Douglass  _et al._ (1993) and bistable single neuron models showing a
correlated switch between the states when they are driven by noise in the
presence of periodic external modulation Bulsara  _et al._ (1991). For SR to
be effective at this scale, the noise must be optimized and adjusted to the
nature of the signal Wiesenfeld  _et al._ (1994).

Stochastic resonance (SR) plays a significant role in ion channel signal
transduction at the subcellular level. This was first demonstrated in a large
parallel ensemble of an artificial ion channel (polypeptide alamethicin
incorporated into planar lipid bilayers), where the signal transduction
induced by noise was increased to over a hundred-fold Bezrukov and Vodyanoy
(1995). Ion channel conformational changes are considered as thermally-
activated Poisson events Bezrukov and Vodyanoy (1997). Ion channels open and
close stochastically and have no built-in “absolute” threshold. However,
additive noise changes their activation barriers linearly with applied
transmembrane voltage, increasing the instantaneous rate of the thermally
activated reaction and exhibiting SR Bezrukov and Vodyanoy (1998). This allows
for the detection of small amplitude input signals in thermally driven
physico-chemical systems where reaction rates are controlled by activation
barriers (such as semiconductor p–n junctions and voltage-dependent ion
channels) Bezrukov and Vodyanoy (1997). However, artificial channels used in
these studies are less sensitive to temperature variations than natural
channels Schmid, Goychuk, and Hänggi (2001); Parc, Koh, and Sung (2009). The
higher temperature sensitivity of natural channels near physiological
temperature (and the induced conformational transitions) effectively changes
the energy landscape and transition rate between closed and open states at the
level of single ion channels Parc, Koh, and Sung (2009). As a result, at the
scale of a single ion channel, enhanced thermal-noise-induced SR affects the
rate of information gain more in biological ion channels much more than in
artificial channels Goychuk and Hänggi (2000); Parc, Koh, and Sung (2009).

The conventional periodic, aperiodic, and nonstationary stochastic resonance
that occur at physiological temperature allows for the harnessing of noise to
enhance signal transmission and information gain with fewer channels Goychuk
and Hänggi (2000); Adair (2003). Harnessing noise through stochastic resonance
is an energy-efficient strategy for cells, as increasing the density of ion
channels to combat noise is very energy-consuming. Additionally, collective
property of ion channels and its effect on SR also controls the response to
noise. The number of ion channels determines the amplitude of the membrane
potential fluctuation Toral, Mirasso, and Gunton (2003). Therefore, in small
clusters of ion channels, SR and threshold-crossing are mostly defined by
individual channel kinetics and the thermal-noise-induced SR Schmid, Goychuk,
and Hänggi (2001). As the size of the membrane patch grows (in soma vs
dendrites for example) and/or density of ion channel increases (via modulation
of gene expression or their polarization as in the case of neurons), a system-
size resonance occurs Pikovsky, Zaikin, and de la Casa (2002). This emergent
collective properties of globally coupled ion channel assemblies exhibits a
resonant-like temporal coherence Schmid, Goychuk, and Hänggi (2001). This
collective property leads to a type of SR that occurs only in large clusters
of ion channels even in the presence of the suboptimal intrinsic noise of
single channels Schmid, Goychuk, and Hänggi (2001); Schmid  _et al._ (2004).
Unlike man-made engineering systems, where efforts are focused on eliminating
noise, cells harness boundary noise at subcellular scale. SR plays a key role
in dealing with boundary noise at both the individual ion channel and
collective levels.

By focusing on the problems cells must solve, we can better understand the
computational primitives operated through cellular biophysical interfaces and
biochemical pathways. The stochastic nature of ion channels and the presence
of stochastic resonance can guide us to probe information gating through the
lens of quantization error. Ligand receptors and ion channels act as
quantizing sampling devices, with their stochastic kinetics and density
defining the collective sampling rate. We discussed how stochastic resonance
can enhance the sensitivity of ion channels to subthreshold inputs. Equally
important, thresholded systems that display SR are special forms of dithered
quantizers Wannamaker, Lipshitz, and Vanderkooy (2000). This property of SR
enables the cells to modulate the sampling precision and information transfer
of ion channels through a quantization operation. Dithering is a common method
in digital signal processing used in domains such as audio and picture coding.
Addition of random noise (or dither) to data prior to storage or transmission,
randomizes the quantization error when a reduction in precision is necessary
Roberts (1962). During WWII, engineers noticed that mechanical computers used
for radars and trajectory calculations operated better onboard airplanes than
on the ground. They realized that vibration-induced noise randomized the
quantization error, increasing the accuracy of the calculations operated by
those mechanical computers Pohlmann (1989). Since then, it has become common
practice to add a small amount of noise to a signal either before quantization
(during analog-to-digital conversion) or when reducing the bit-depth of a
digital signal (such as during a 256 bit to a 16 bit image reduction).

Thresholded dynamical systems such as ion channels and excitable membranes can
be viewed as multistable systems. Signal transduction through the channel has
a finite range and sampling occurs in a discrete stochastic fashion, resulting
in inevitable distortion and loss of signal details. This distortion and loss
of signal details can take the form of spurious frequencies or amplitude
reduction due to time discretization and amplitude quantization. In systems
with a nonlinear input-output relationship, minimizing distortion requires the
quantization error to be independent of the input signal Gammaitoni (1995a).
In thresholded nonlinear dynamical systems, such as cells with excitable
membranes, stochastic resonance is more like a special case of a dithering
effect rather than a resonant phenomenon Wannamaker, Lipshitz, and Vanderkooy
(2000); Gammaitoni (1995a). Just as image dithering preserves the gradient of
color-depth during bit reduction, for example for converting gray-scale to
black-and-white, SR in multi-thresholded dynamical systems can increase the
number of quantization levels Gammaitoni (1995b). As the SR affects the
responsiveness of individual channels to stochastically open at a different
rate, collectively the opening dynamics create a multi-threshold system.
Instead of setting a fixed threshold for all channels to suddenly switch to
the open state, presence of noise induces concentrated transitions to open
state in clusters of channels. The small intervals created around multiple
thresholds, created by different sizes of these clusters, increases the number
of quantization levels. This increased quantization level enables the cell to
transduce microscopic fluctuations in the external environment in a graded
discretized fashion. As a result, by harnessing the noise, cells effectively
capture a macroscopic view of their environment at the boundary. We can deduce
that SR enables the cells to be sensitive to microscopic fluctuations while
maintaining a macroscopic grasp of their environment.

The ability of cells to operate efficiently through noisy dynamics highlights
the functional role of noise in biological systems Azpeitia, Balanzario, and
Wagner (2020); Tsimring (2014). The fact noise at different levels can have
non-rigid functional characteristics, such as boundary stochastic noise in
single channel, cluster of channels in a membrane patch, collective properties
of polarized distribution of channels in a single cell, and in network of
cells, underscores the importance of multi-level modeling. The multiscale
effects of stochastic resonance fits well with the notion that there is no
privileged level of causality in biological systems Noble and Noble (2018).

## Conclusion

Hilbert’s list of problems was intended not just as a set of unsolved
mathematical problems, but as a framework for advancing the discipline. While
some of these problems remain unsolved, the idea of a framework has endured.
Unlike mathematics, which is based on logic and proofs, biology has mostly
been an ever-growing collection of detailed findings with limited scope.
Attempts at biological modeling and theorizing have had limited success. What
are we missing? Is biology fundamentally different from other fields such as
physics? Can it not be tamed with mathematical models? In a thoughtful essay
Gunawardena (2014), the author argues that, at least at the molecular scale,
our models are mostly phenomenological and not much more than an educated
guesswork. Gunawardena mentions that unlike physics, biological models are not
objective and therefore our best chance is to start with a good choice of
assumptions that can guide us to choose the right flavor of the model that
would suit the problem Gunawardena (2014). The essay paraphrase James Black
(Noble laureate pharmacologist) and reminds us that “our models are accurate
descriptions of our pathetic thinking”. This portrayal of biological models
points to key issues. However, an often unspoken problem with biological
models is that no matter how good they may be, the disconnect between the
models prevents us from constructing fundamental theories. As a result,
biological sciences have grown exponentially in data size but made minimal
advances in principles. Ironically, if there are biological textbooks with
phrases like “fundamentals” or “principles” in the title, with each edition
the textbook grows only by volume without ever delivering the promising title.
This work aims to tackle this challenge by proposing a program inspired by
Hilbert’s approach.

To bring different models together, it is important to focus on problems
rather than methods. In sketching the problems that biological systems have to
solve, constraints become vividly apparent. Problems and constraints will help
us to find the path that will keep us true to data, force our models to be
falsifiable, and will give us an underpinning pathway for connecting the
models and searching for general principles. Adhering to constraints and
details rooted in the physical nature of biological components, will give us a
chance to formulate models that are true to the underlying biophysics with
predictions that can be experimentally falsified Bialek (2012). It is
essential to avoid getting trapped by isolated reductionist descriptions of
biological systems. A computational lens can help us focus on the function of
a system rather than its individual components. There are convergent themes
between biological and computing systems that fall under the lens of
“computational thinking”, such as distributed and coordinated processes,
robustness to failure/attacks and modularity Navlakha and Bar-Joseph (2011).
The proposed framework aims to combine computational lens and biophysical
modeling. An important outcome is the ability to design experiments with
search for principles in mind. The ability to adopt a reductionist physics
perspective within the umbrella of a functional decomposition also separates
our framework from that of cybernetics. Cybernetics correctly recognized the
crucial importance of feedback and control in biological systems Wiener
(1965). However, cybernetics’ overemphasis on teleology Rosenblueth, Wiener,
and Bigelow (1943) juxtaposed it with the explosive success of reductionist
molecular biology. While one can argue that what makes biology truly unique is
its concern with purpose Mayr (2004), perhaps searching for behavioral
purposeful descriptions untied from the underlying biophysical processes was a
key behind cybernetics’ demise. We can bring back the insightful aspects of
cybernetics to the center stage and avoid its pitfalls.

Models that aim solely to fit data better than their competition are non-
falsifiable Gunawardena (2014). While artificial neural networks (ANNs) have
been successful in modeling complex pathways of communication and control,
their success is rooted in a nonlinear fitting process. This can result in
replacing one complex nonlinear system with another black box that we don’t
understand. A better approach may be to use ANNs as nonlinear optimizers
within mathematical models Rackauckas  _et al._ (2020); Karniadakis  _et al._
(2021). This can fuse physical laws with data-driven approaches to describe
and predict system behavior and test the falsifiability of our theories.
Advances in machine learning can also help combine data-driven methods with
physical principles to discover accurate models of complex dynamical systems
from noisy and incomplete data Lu, Ariño Bernad, and Soljačić (2022); Reinbold
_et al._ (2021). A framework that can bind problems and pave the way for using
these mixed models to search for principles could be a major leap forward for
biology. With advances in precision devices, high-throughput techniques,
experimentation scale-up, computing hardware/software and AI, we are at a
prime point to make significant progress. Theoretical frameworks for cellular
computation and cell-cell communication may provide the foundation for biology
to develop its first principled quantitative theories in the 21st century,
similar to the impressive achievements of physics in the early 20th century.

###### Acknowledgements.

The author would like to express gratitude to the organizers and attendees of
the “What is biological computation?” workshop at Santa Fe Institute. Some of
the ideas presented in this work were inspired by discussions that took place
during the workshop.

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