# Biological efficiency in processing information

Dorje C. Brody1 and Anthony J. Trewavas2 1School of Mathematics and Physics,
University of Surrey, Guildford GU2 7XH, UK  2Institute of Molecular Plant
Science, University of Edinburgh, Edinburgh, UK

###### Abstract

Signal transduction, or signal-processing capability, is a fundamental
property of nature that manifests universally across systems of different
scales – from quantum behaviour to the biological. This includes the detection
of environmental cues, particularly relevant to behaviours of both quantum
systems and green plants, where there is neither an agent purposely
transmitting the signal nor a purposefully built communication channel. To
characterise the dynamical behaviours of such systems driven by signal
detection followed by transduction, and thus to predict future statistics, it
suffices to model the flow of information. This, in turn, provides estimates
for the quantity of information processed by the system. The efficiency of
biological computation can then be inferred by measuring energy consumption
and subsequent heat production.

## Introduction

Plants in wild circumstances commonly experience a fluctuating environment
that can threaten the survival of any individual and thus limit its capability
to reproduce. Within reasonable environmental limits, plants react to such
variations by changing their behaviour adaptively, improving their probability
of survival Dobzhansky . They detect and transduce the information they
receive from their environment and use that processed information to establish
counter-reactions in behaviour to improve survival. Unlike animals that can
move out of danger, plants in general cannot move away from extreme threats
except via distribution of seed or pollen or the production of dormant
underground organs such as tubers, fleshy roots and stolons that weather the
hazard and produce new shoots when conditions change. When milder threats are
experienced from numerous biological and abiological threats, developing
plants counteract by changing development (phenotypic plasticity) or molecular
physiology to improve the probability of survival Gilroy . In 1871, Charles
Darwin Dawin  commented that “intelligence is based on how efficient a species
becomes at doing the things they need to survive,” leading to the general
recognition that “adaptive behaviour is the essence of (biological)
intelligence” Anastasi ; Beer . Plants use the information that they receive
from their noisy environment to construct adaptive responses. Those species
that do so more efficiently may benefit in the struggle for existence. But in
establishing the reality of Darwin’s view, there is a necessity to quantify
the environmental information that any plant experiences and in turn be able
to measure how efficiently it is processed. It is obvious that all plants
experience information from their environment and that is commonly stated, but
without attempts to quantify what is experienced, the term really has little
meaning. Many areas of plant science would benefit if this can be achieved as
would understanding of biological intelligence.

Intensive research in the last 30 years has identified the central role of
cytosolic calcium $[{\rm Ca}^{2+}]_{\rm i}$ in transducing most environmental
signals experienced by plants Gilroy ; Kudla . Signals are perceived by
specific receptors often located in the plasma membrane. Occupied receptors,
it is assumed, form membrane patches that initiate intracellular calcium
transients, $[{\rm Ca}^{2+}]_{\rm i}$, within less than a second and lasting
at the very most a minute. Inhibition of these transients prevents the
subsequent physiological or phenotypic changes that normally occur hours, or
even days or weeks later. These transients can oscillate if the environmental
change is prolonged Gilroy ; Kudla ; Delormel . Cytoplasmic changes in $[{\rm
Ca}^{2+}]_{\rm i}$ are perceived by well over 100 proteins (CDPKs, calcium-
activated kinases, calcium-calmodulin kinases, calmodulin, calmodulin-like
proteins, CBL kinases, calcineurin-like proteins) that bind ${\rm Ca}^{2+}$
directly. A variety of channels, pumps, exchangers and carriers are
responsible for ${\rm Ca}^{2+}$ entry. These membrane-bound proteins help
shape and change the internal spatial distribution generating a distinctive
signature often characteristic of the specific signal. It is thought that
different signals recruit different combinations of these ${\rm Ca}^{2+}$
binding proteins that continue information transfer into hundreds of protein
kinases of differing kinds that manipulate further downstream change into
altered ion fluxes, enzyme activities, cytoskeletal reconstructions, and
ultimately particular combinations of gene expressions.

Whatever the mechanism by which information is detected, transmitted,
processed, stored, and in some cases erased, the amount of information
processed can be estimated from dynamical behaviours resulting from the
processing of information, and the efficiency of biological computation can be
inferred from the amount of energy consumed by the cells to process
information. That is, these estimates can be made without necessarily the
knowledge of detailed structures and mechanisms for achieving the detection of
environmental conditions by plants Diaz . The purpose of the present paper is
to explore this line of investigation.

## Information in wild environments will be noisy

From the point of view of mathematical modelling of adaptive behaviour, what
underpins adaptation is the detection and processing of noisy information
about environmental conditions. That is, adaptation to changes of external
conditions is not possible without the information about those noisy
conditions. The processing of information, in turn, leads to adaptive
dynamics, because the state of a biological system can be viewed as the
reflection of the system’s perception of the condition of its environment.
Thus, we are led to the hypothesis that dynamical behaviours of all biological
systems originate in signal detection, and that by modelling the noisy
information flow for environmental cues, it is possible to predict the
statistics of biological dynamics by means of a logical deduction based on
optimal (efficient) signal detection.

The information-based approach to modelling biological dynamics allows for a
way of measuring information contents of environmental cues, or messages, that
biological systems such as green plants experience. When environmental
conditions change, much of the processed information about previous conditions
will presumably be erased. For instance, when a climbing plant detects an
object that can support its growth, but then the object is removed, it seems
likely that this information is erased. Otherwise, plant cells will require
considerable memory capacity to store all the day-to-day processed
information, which seems unlikely. Of course, if erasure of processed
information results in energetic penalties, for instance the leaf-folding
behaviour of Mimosa pudica under false alarms, then there is an advantage in
storing processed information, but even here the memory appears to be erased
within thirty days GRDM . However, if such processed information is regularly
erased, then as a consequence we can estimate the computational efficiencies
of biological systems. This is because information erasure requires energy
consumption indicated by additional heat production, and this can be
calculated and potentially measured, and compared to the total energy
consumption and heat production of biological cells.

## Such adaptive dynamics also has relevance at the quantum level

Perhaps unexpectedly, this adaptive dynamics appears to have a deeper,
quantum-mechanical origin. Consider a quantum system, like a particle, in an
external field, say, a photon field. For the state (wave function) of the
system to evolve in time in accordance with the conditions of the external
field, it is required to extract information about the condition of the
environment. This information is necessarily noisy, but if we assume that
nature is intrinsically efficient in the sense that dynamical evolution is
dictated by the criterion of minimising the degree of uncertainties about the
condition of the environment, then the system must follow the dynamics that is
consistent with optimal signal detection. Such a picture for the dynamics of a
quantum system is not entirely dissimilar to, say, a plant cell extracting the
information of the light source from noisy interaction with a photon field for
the phototropic or heliotropic dynamics. Indeed, there is a great deal of
parallelism between quantum and biological dynamics (not to be confused with
static aspects such as quantum coherence or entanglement that may or may not
be of importance in biology), and this can be exploited to pave the way
towards a better understanding of information and signal transduction in
biology, because the theory of quantum dynamics in noisy environments is well
developed, while biological dynamics is less understood.

But how is such a characterisation of quantum dynamics as extracting
information from the environments compatible with the more traditional
formulation of quantum systems by means of Schrödinger’s wave equation? And
can we show that nature is indeed efficient in its ability to process
information? If so, what are the implications in biology, if any, and what do
we learn from these observations?

The purpose here is to clarify some of these issues, with the view towards
building a unified information-based approach to modelling quantum and
biological systems, and in particular, plant dynamics. Specifically, we begin
by explaining how the specification of the information flow can be used to
model dynamic features of biological systems. In particular, focusing on the
response of green plants to environmental cues, it is shown how the techniques
of information and communication theory can be used to characterise their
dynamics, even though in the context of signals from environments there are no
agents purposely transmitting these signals. The sun does not emit photons
with the purpose of letting green plants on earth know of its whereabouts.
Wind does not blow at high speeds for the purpose of stressing plant
structures. The environments, in effect, carry the signals, but they are not
purposefully built like a telephone line carrying signals.

We then show that precisely the same mechanism of signal-detection-induced
dynamics is already present at the quantum-mechanical level. After
illustrating how the orthodox views of quantum theory requires a closer re-
examination, taking into account the impact of environments – more precisely,
of noise – we show how the modified formulation of quantum theory is
underpinned by optimal signal detection problems. In particular, it is shown
how signal detection leads to the so-called decoherence effects of quantum
mechanics. We then discuss the advantages of applying models describing
dynamics of quantum systems for the characterisation of biological dynamics,
when there is an incompatibility between adaptation and the rate of change in
the environment. Such dynamical models not only allow for the prediction of
future statistics but also enables the quantification of information contents
that are otherwise difficult to estimate from experimental data. We conclude
with some discussion on open challenges in biology from an information-based
perspective.

## Processing of noisy information induces dynamics in biology

Information theory is traditionally concerned with the amount of information
contained in a signal transmitted through a noisy channel, the informational
capacity of the channel, and corrections of errors Kullback ; Cover . These
ideas refer to an established communication channel used by a sender and a
receiver of the message, which, in biological contexts, might translate into
signal transduction in molecular and cellular networks Cheong .

As well as internal communication channels, biological systems constantly
extract information from their environments, in the absence of a transmitter
of a message or a communication channel designed to deliver specific
environmental cues Mescher . Take, for instance, the case of green plants.
Whether it is sunlight, scattered photons, volatiles, wind, water, soil
nutrients, or gravity, information about these environmental cues is processed
effectively Trewavas . Even when there are well-defined communication
channels, estimating the amount of information transmitted from data is not
straightforward Thomas , so the problem becomes even less tangible when it
comes to quantifying information processed by plants through what seems to be
ambiguous communication channels. This is where a mathematical model can be
exploited to better understand the link between information processing and
resulting biological dynamics.

Processing of information commonly leads to phenotypic changes and movements
in plants. For instance, leaf blade reorientation towards the direction of the
sun. Such a dynamical behaviour is induced by detection and processing of
signals encoded in photons. Now in the conventional approach to mathematical
modelling of biological dynamics, one typically attempts to model the
resulting motion directly in the form of differential equations, which in some
cases may be superimposed with additional noise terms. While such an approach
of directly modelling observed dynamics has seen some successes in modelling
certain specific systems, for, models can always overfit to match simple
dynamics, ultimately it appears futile to attempt to directly model dynamical
behaviours of biological systems driven by processing of noisy information.

The reason is a simple one. What underlies biological dynamics is the noisy
information about environmental conditions, which are processed to arrive at
choices, and hence dynamics. The input, therefore, is the specification of
noisy information, on which optimal signal detection can be applied to arrive
at the output, and this leads to phenotypic change and/or movement. This
input-output transformation is typically highly nonlinear. Hence any attempt
to model the output will fail to capture the input-output causal relation. In
other words, direct modelling of the dynamics in most cases cannot replicate
the underlying information-processing features.

The way forward, therefore, is to model the noisy flow of information, from
which the implied dynamical behaviour of a given biological system can be
deduced (rather than modelled) using the theory of optimal signal detection.
In this way the causal structure underlying the resulting biological dynamics
becomes transparent.

## Environment acting as a communication channel

Of course, in many cases there are no purposefully built communication
channels when it comes to biological systems processing information about
environmental cues, and this fact has perhaps prevented the application of
communication theory to describe biological dynamics. To proceed, therefore,
we must regard the environment as a whole, in effect, as playing the role of a
communication channel. The idea is as follows. Take the case of a green plant.
There is a range of quantities of interest that plants wish to identify, such
as the location of the light source, the direction of the gravitational pull,
the gradient of volatile concentration in the air or soil, the existence of
plants or other objects in their vicinities, and so on Trewavas2 . We may
refer to these quantities as ‘signals’ or ‘messages’, even though in none of
these cases, an active agent is transmitting the signal. Yet, these signals do
exist, not in abstraction but in reality, and they are carried by the
environment (air molecules, photons, gravitons, and so on) to reach, along
with noise (e.g., scattered photons), the plant receptors.

From the viewpoint of communication theory, whether there are active
transmitters of signals or purposefully built channels is immaterial, because
information has been successfully transmitted, albeit superimposed with noise.
It is the environment itself that generates what plants perceive as signals,
and it is the environment itself that carries these signals along with noise.

## Dynamics following from Bayesian updating

In many cases, the signal (i.e. the quantity of interest from the plant’s
perspective) is given by a time series, and so is noise, but for an
illustrative purpose take for simplicity the case in which they are fixed in
time. In more specific terms, let $X$ be the random variable representing the
signal, and suppose that it can take $N$ discrete values $\\{x_{i}\\}$ with
the a priori probabilities $\\{p_{i}\\}$ (the continuous case can equally be
treated). Similarly, let $\epsilon$ represent noise, with the density function
$f(x)$. Information acquisition in signal detection is then modelled by
learning the value $\xi=X+\epsilon$ of signal plus noise. The motion of plant
(or any other biological system for that matter) resulting from information
acquisition will then be consistent with the transformation from the a priori
mean of $X$ to its a posteriori mean Wiener :

$\sum_{i=1}^{N}x_{i}p_{i}~{}\Rightarrow~{}\sum_{i=1}^{N}x_{i}\left(\frac{p_{i}\,f(\xi-
x_{i})}{\sum_{j=1}^{N}p_{j}\,f(\xi-x_{j})}\right).$

Here, the term given in the bracket on the right side is just the Bayes
formula for the a posteriori probability of $X$, subject to the specification
of partial information $\xi$ about $X$.

To gain a better intuitive understanding of the information-based formulation,
consider the experiment on the parasitic plant Cuscuta pentagona (dodder) to
demonstrate its ability to detect the host location via inferring the gradient
of volatile chemical concentration in the air DeMoraes . The initial motion of
the seedling as circumnutation is dictated by the genetic information already
encoded in the plant (as opposed to processed information), and suggests that
the a priori distribution of the host location is uniformly distributed over a
circle centred at the point of germination, on account of the symmetry of the
configuration. But then the detection of volatiles in the course of nutation
turns the a priori distribution into the a posteriori distribution, resulting
in the observed directed growth orientation towards the host location DeMoraes
. The a posteriori distribution can be estimated accurately by increasing the
sample size considered in DeMoraes . The amount of information processed can
then be identified, in units of bits, by working out the relative entropy or
entropy reduction associated with the a priori and the a posteriori
distributions.

This example illustrates how a seemingly abstract mathematical idea of signal
detection can be used to characterise the dynamical behaviours of biological
systems. This quintessentially biological feature of ‘intelligently’ adapting
to environmental conditions, in the sense of optimally processing noisy
information so as to reduce, on average, the degree of uncertainty as measured
by entropy, however, seems to have a deeper quantum-mechanical origin. That
is, dynamical features of systems induced by signal transduction appears to be
universal across all scales.

## Role of environmental noise in quantum dynamics

In physical science the dynamics of a system is determined by the
specification of its energy. That is, once the system energy is specified,
then equations of motion for the state of the system unambiguously determine
how the initial state evolves in time. From Newtonian mechanics to quantum
theory, this approach has proven incredibly effective. Quantum theory, in
particular, is one of the most stringently tested theories in science,
enabling high-accuracy predictions for experimental data. Yet, there are
reasons to doubt whether this energy-based description of physical systems,
supplemented with an absolute determinism, is complete in the quantum realm.
What lies at the heart of the issue is an internal contradiction in the
conventional characterisation of a quantum system. Namely, a quantum system in
isolation is defined to be one that is modelled by its Hamiltonian (energy),
which determines the system dynamics according to the Schrödinger equation. In
most example systems, however, their Hamiltonians represent system-environment
interactions, such as the effects of external fields; but this contradicts the
assumption that the system is in isolation.

From the perspective of a quantum system, the interaction with its
environment, more specifically, is interaction with environmental particles
such as photon fields, and this is inadvertently noisy owing to the existence
of the very large number of particles involved. The omission of the influence
of noise in the theoretical description (such as Schrödinger’s theory) in turn
leads to one of the central issues in contemporary physics of explaining the
transition from quantum to classical, as exemplified by the so-called
measurement problem of quantum theory Isham . For sure if a system is in
equilibrium with its environment, then it is expected that the system energy
is conserved on average subject to fluctuations due to noise, and that the
degree of fluctuation scales down in the limit whereby the energy gaps of the
system are sufficiently small (cf. Adler ). This leads to the hypothesis that
one should regard Schrödinger’s theory as an approximation to a more realistic
description of the quantum world in which the role of noise is taken on board.

Of course, any deviation from the noise-free deterministic ‘unitary’ dynamics
(if the relative separation or difference of two wave functions remains
constant in time, then the dynamics is called unitary) predicted by
Schrödinger’s theory must be subject to scrutiny. This is because if the
Schrödinger equation for the wave function of the system is replaced with a
stochastic dynamical equation that reflects the impact of noise, then
experimental predictions will in general deviate away. However, for a number
of prototypical quantum systems for which ample experimental data exists, this
deviation has been shown to fall within error bars, and hence not detectable,
provided that the random dynamics of the wave function conserves energy
expectation Adler . In other words, there is a family of stochastic models
describing the dynamics of quantum systems that does not contradict the
successful predictions arising from the traditional quantum theory, while
incorporating the impact of noise.

This may seem surprising, but the intuitive reason is that the timescale for
such a stochastic dynamics to deviate significantly away from unitarity is
inverse proportional to system’s energy variance H . Hence for a small quantum
system, for which the energy variance is small, the stochastic evolution will
maintain approximate quantum coherence for long times. On the other hand, if
the system is coupled to a measurement apparatus, then the energy variance is
amplified by the uncertainties of the macroscopic pointer state (the state of
the pointer of the apparatus pointing in one direction or another will result
in huge energy uncertainty), forcing the wave function to ‘collapse’ into one
of the stable eigenstates instantly.

## Characteristics of quantum dynamics induced by signal detection

What then is the main trait of such stochastic models? It can be shown that
for the class of stochastic Schrödinger equations for which the energy
expectation is conserved, their solutions can be obtained by formulating
underlying signal detection problems about the energy of the system BH1 ; BH2
. Suppose that a quantum system (say, a particle) is immersed in an external
field, and the interaction of the two is modelled by a Hamiltonian
${\hat{H}}$, which characterises the energy of the system. One can think of
${\hat{H}}$ as a matrix with eigenvalues $\\{E_{i}\\}$, representing definite
energy values. An energetically definite (certain) state of the system is
represented by the eigenvector $|E_{i}\rangle$ of ${\hat{H}}$ corresponding to
the eigenvalue $E_{i}$. A generic state $|\Psi\rangle$ of the system can be
expanded in terms of the eigenvectors of the Hamiltonian:

$|\Psi\rangle=\sum_{i}\sqrt{p_{i}}\,|E_{i}\rangle.$

Here, we have ignored the phase factors that are quantum-mechanically
important but not relevant to the ensuing discussion. The expansion
coefficients $\\{\\!\sqrt{p_{i}}\,\\}$ are such that their squares add up to
one. These are then interpreted as determining the probability of finding the
system to be in the state with a definite energy. Such a generic state is
therefore a state of indefinite energy. Our hypothesis then is that through
interaction with the environment, however, the system acquires partial
information about energy – partial because of the prevalence of noise.

Typically the information acquisition occurs continuously in time, but for
simplicity consider a ‘single-shot’ transfer of information. The (unknown)
quantity of interest is the energy, which we model by a random variable $H$
taking the value $E_{i}$ with the probability $p_{i}$. We let $\epsilon$ model
the noise, which is assumed to be independent of the ‘signal’ $H$ and has the
density $f(x)$. Then information transfer from the environment to the system
can be modelled by the specification of the value $\xi=H+\epsilon$ of the sum
of signal and noise. There are two unknowns, $H$ and $\epsilon$, and one
known, $\xi$, so this data is insufficient to determine the value of $H$, but
it can be used to reduce the uncertainty of $H$, just as in the biological
case. Such a reduction results in updating the prior probabilities
$\\{p_{i}\\}$ into posterior probabilities $\\{\pi_{i}\\}$. If this updating
of information is efficient in that it maximally reduces the uncertainty, then
$\pi_{i}$ must be the probability that $H=E_{i}$ conditional on the value of
$\xi$, and this can be worked out by use of the above-mentioned Bayes formula.
Hence the state of the system, after acquiring this information, and if nature
is efficient, is given by

$|\Psi^{\prime}\rangle=\sum_{i}\sqrt{\pi_{i}}\,|E_{i}\rangle,$

where the conditional probability $\pi_{i}=\pi_{i}(\xi)$ is a function of the
information transmitted. The acquisition of information by the system,
represented by the transformation $p_{i}\to\pi_{i}$, then results in the well-
known decoherence effect of quantum theory, as described below.

## Acquisition of noisy information leads to decoherence effects

From the perspective of an external observer (an experimentalist), it is
useful to represent the state of the system in the form of a density matrix
${\hat{\rho}}=|\Psi\rangle\langle\Psi|$, where $\langle\Psi|$ is the complex
conjugate transposition of $|\Psi\rangle$. This is because the expectation of
a physical observable, represented by a matrix ${\hat{F}}$, in the state
$|\Psi\rangle$ is given by $\langle\Psi|{\hat{F}}|\Psi\rangle={\rm
tr}({\hat{\rho}}{\hat{F}})$, and this is the quantity that would be measured
in laboratories. If the observer knows the state of the system, then writing
the expectation in terms of a pure state $|\Psi\rangle$ or in terms of a
density matrix ${\hat{\rho}}=|\Psi\rangle\langle\Psi|$ makes no difference.
However, if the observer is uncertain about the state of the system, and the
uncertainty being represented by a distribution over pure states, then we need
to average not the wave function $|\Psi\rangle$ but the matrix
$|\Psi\rangle\langle\Psi|$ using this distribution, and the resulting averaged
matrix is what defines the density matrix. This follows on account of the fact
that observable quantities are not linear but quadratic in the wave function,
and the square of the mean is not the same as the mean of the square.

In the energy basis, the initial state ${\hat{\rho}}$ thus has the matrix
elements $\sqrt{p_{i}p_{j}}$, which transform into
$\sqrt{\pi_{i}(\xi)\pi_{j}(\xi)}$ after information acquisition. The external
observer, however, has no information about the value of $\xi$, so at best
they may consider the average
${\mathbb{E}}[\\!\sqrt{\pi_{i}(\xi)\pi_{j}(\xi)}]$ over all values $\xi$ can
take, and this gives the matrix elements of the state
${\hat{\rho}}^{\prime}={\mathbb{E}}[|\Psi\rangle\langle\Psi|]$ as represented
by the observer after the information acquisition by the system. Here,
${\mathbb{E}}[-]$ denotes expectation over the random variable $\xi$, which in
the present example has the density $p(\xi)=\sum_{i}p_{i}f(\xi-x_{i})$. Then
from the Bayes formula, a calculation shows that the transformation of the
matrix elements takes the form
$\sqrt{p_{i}p_{j}}\to{\mathbb{E}}[\\!\sqrt{\pi_{i}(\xi)\pi_{j}(\xi)}]=\sqrt{p_{i}p_{j}}\Lambda_{ij}$.
Here, the symmetric matrix $\Lambda_{ij}$ is given by

$\Lambda_{ij}=\int\\!\sqrt{f(\xi)\,f(\xi+\omega_{ij})}\,\mbox{$\rm d$}\xi,$

where $\omega_{ij}=E_{i}-E_{j}$ and $f(x)$ is the density for $\epsilon$.
Evidently, we have $\Lambda_{ii}=1$ for all $i$ and $\Lambda_{ij}<1$ for
$i\neq j$ (assuming that $E_{i}\neq E_{j}$). Therefore, acquisition of
information by the system will result in the damping of the off-diagonal
elements of the density matrix. This is the so-called decoherence effect. In a
typical (but not always the) case, the damping factor $\Lambda_{ij}$ of the
off-diagonal element is more pronounced for wider energy gaps $\omega_{ij}$.

Evidently, every biological process will exhibit a form of decoherence effect
because an external observer has no access to the precise information acquired
by the biological system, although the effect may not manifest in the form of
a density matrix. The point is that there is a kind of trade-off of
information flow: acquisition of information by the system will result in loss
of information by the external observer. The amount of information gained by
the system is the reduction of the Shannon-Wiener entropy associated with the
transformation $p_{i}\to\pi_{i}(\xi)$, and when averaged over $\xi$ this is
given by $S_{\epsilon}-S_{\xi}$, where $S_{\epsilon}=-\int f(x)\log
f(x)\,\mbox{$\rm d$}x$ is the entropy (measure of uncertainty) of noise and
$S_{\xi}=-\int p(y)\log p(y)\,\mbox{$\rm d$}y$ is the entropy of the
information $\xi$. This follows simply by averaging the entropy change

$\Delta S=-\sum_{i}\pi_{i}(\xi)\log\pi_{i}(\xi)+\sum_{i}p_{i}\log p_{i}.$

In contrast, upon system’s acquisition of information, the observer loses
track of the state of the system, and hence loses information, and this is
measured in terms of the von Neumann entropy, given by $-{\rm
tr}({\hat{\rho}}\ln{\hat{\rho}})$. The value of the von Neumann entropy,
however, does not agree with the change of the Shannon-Wiener entropy, because
the density matrix ${\hat{\rho}}$ does not contain all the distributional
information of the density function $p(\xi)$.

## Quantum evolution is efficient in that it corresponds to optimal signal
detection

More generally, for a broad range of quantum systems immersed in a general
open environment, the state of the system as perceived by an external observer
is described by a density matrix, whose dynamical evolution is described by
the Lindblad equation BP . This equation is deterministic, because the effect
of noise is averaged over all random realisations of the noisy information
when transforming from a wave function to a density matrix. Given this
deterministic equation, finding a randomly-evolving wave function
$|\Psi\rangle$ such that the associated density matrix
${\mathbb{E}}[|\Psi\rangle\langle\Psi|]$ obeys the Lindblad equation is known
as the stochastic unravelling problem. In other words, can we find a
stochastic evolution of the wave function of a system such that when the
effect of stochasticity is averaged over, the dynamics obeys the Lindblad
equation?

It has been demonstrated recently that solutions to the stochastic unravelling
of the Lindblad equation for open quantum dynamics are necessarily those for
the optimal signal processing DCB . That is to say, a noisy evolution of the
wave function that gives rise to the Lindblad equation is one that corresponds
to the optimal detection of the condition of the environment. Given that the
Lindblad equation describes the dynamics of a very wide range of quantum
systems BP , the fact that the equation is underpinned by optimal signal
detection supports our hypothesis that quantum systems evolve so as to on
average maximally reduce the uncertainty of the conditions of the environments
as encoded in the wave function.

The optimality condition here is important, because it establishes the
informational efficiency of quantum dynamics. Hence the feature of an
efficient adaptation to environmental conditions is not unique to biological
systems, and is embedded at the quantum-mechanical level. In fact, we would
argue that this is why biological systems, built out of efficient quantum
particles, are able to efficiently process information.

## From quantum to biological dynamics

The foregoing observations suggest that the dynamical equation governing the
evolution of the wave function of a quantum system is associated with the
optimal signal detection under noisy exchange of information with the
environmental particles – about the conditions of the environment. If so, then
a more realistic description of a quantum system is obtained by the
specification of the noisy information flow concerning the system-environment
interaction, along with the application of optimal signal detection. This then
provides a more complete characterisation of a quantum system, as opposed to
the mere specification of system energy. Putting it differently, because of
the prevalence of noise, what matters ultimately is not the energy but the
information about the energy, or more generally information about the state of
the environment. If this is the case, then the adaptive dynamics of a system,
based on noisy information about the conditions of the environment, is not
unique to biology, but is a universal feature applicable equally to quantum
systems as well as to biological. That is, signal transduction is applicable
universally across all scales. This, in turn, leads to our proposal that
mathematical models used to describe the dynamics of open quantum systems can
be applied, mutatis mutandis, to model the dynamics of biological systems.

Motivated by these hypothesises, stochastic Schrödinger equations have been
applied recently to model heliotropic and gravitropic motions of green plants
DCB . These models have the advantage of predicting the statistics of
biological dynamics. Further, one can use these models to work out the amount
of information required to be processed in order to achieve certain objectives
(for example, for a Sunflower to identify the location of the sun and turn
towards it). While these models are effective in some ways, it is legitimate
to ask whether any of the quantum effects might play a role in the biological
context. The fact that signal detection underpins the dynamics of quantum
systems as well as those of biological systems, in itself, does not imply that
quantum effects manifest themselves in the biological context.

## Manifestation of a quantum effect in biology

To better understand the role that quantum features might play in biology and
ecology, we remark first that the stochastic Schrödinger equations are
specified by two drivers, namely, the Hamiltonian ${\hat{H}}$ and the Lindblad
operator ${\hat{L}}$. Now one of the fundamental features of quantum theory is
that observable quantities need not be compatible. So for instance while in
classical physics a particle will always have a definite position and a
definite momentum, in quantum theory there exists no state (wave function) of
a particle such that its position and momentum take definite values (i.e. with
no uncertainty) simultaneously. In the case of a stochastic Schrödinger
equation, we have the operators ${\hat{H}}$ and ${\hat{L}}$. They can either
be compatible so that ${\hat{H}}{\hat{L}}={\hat{L}}{\hat{H}}$ (the matrix
product is commutative), or incompatible so that
${\hat{H}}{\hat{L}}\neq{\hat{L}}{\hat{H}}$ (matrices do not commute). In the
latter case, no state can take definite values for both ${\hat{H}}$ and
${\hat{L}}$.

When a stochastic Schrödinger equation is applied in the biological context,
the Hamiltonian ${\hat{H}}$ models the change of environmental conditions
itself, while the Lindblad operator ${\hat{L}}$ models adaptation in response
to environmental cues. It is more often the case that these two effects are
not compatible; biological systems are capable of adaptation so long as the
environmental conditions do not change too abruptly. Otherwise, a rapid change
of conditions can be catastrophic for survival. Hence there are two phases,
one phase in which adaptation is feasible, and another phase in which
adaptation is impossible. Such an effect can be modelled with a stochastic
Schrödinger equation for which, and only for which ${\hat{H}}$ and ${\hat{L}}$
are incompatible. That is, depending on the relative magnitudes of ${\hat{H}}$
and ${\hat{L}}$, the dynamics of the system exhibits different
characteristics. Although detailed features of such phase transitions have not
been fully uncovered, their existence is well documented Bassi ; BL .

Therefore, while from a purely signal-detection perspective there is little
evidence that some of the counter-intuitive quantum features (such as
entanglement or phase coherence) play any role in biology and ecology, the
lack of compatibility among observable quantities – another quantum feature –
does appear to play an important role. This observation makes the stochastic
Schrödinger equation a useful and effective mathematical tool for
characterising biological dynamics, even though in many cases the dynamics can
equally be modelled using the language of classical signal detection.

## Quantifying information contained in environmental cues

Whether classical or quantum signalling models are used to characterise
biological dynamics, the use of these models has the advantage – apart from
being able to predict the statistics of future events – of allowing for the
quantification of information. This is of particular interest in the case of
plants, for, we do not normally associate the notion of ‘information
processing’ to plants, and yet, plants do process environmental information at
all times. Thus, models can be used to assist estimating the information
contents of environmental cues. Take, for instance, the example considered
earlier in which the signal is modelled by the random variable $X$, taking the
value $x_{i}$ with the probability $p_{i}$, and noise by $\epsilon$, with the
density $f(x)$. Then the information contained in the noisy observation
$\xi=X+\epsilon$ about the value of $X$ is measured by the mutual information
$J(\xi,X)$. This is the amount of information contained in environmental cues
concerning the signal of interest for the plant. Writing
$p(\xi)=\sum_{i}p_{i}f(\xi-x_{i})$ for the density of $\xi$, the mutual
information can be shown to be given by the entropy difference GY ; BDFH

$J(\xi,X)=\int f(x)\log f(x)\,\mbox{$\rm d$}x-\int p(\xi)\log
p(\xi)\,\mbox{$\rm d$}\xi.$

Remarkably, this agrees with the negative of the average change of entropy
obtained earlier. Therefore, once a model for the signal and noise is chosen,
one can predict the statistics of biological dynamics by using the a
posteriori probabilities. Model parameters can then be calibrated by comparing
the statistics thus obtained against empirical data. When a model is
calibrated, it can be used to quantify the information content. In other
words, even though there is neither an intentional transmitter of the signal
nor a purposefully built communication channel, it is nevertheless possible to
estimate the information content of environmental cues.

## Discussion

The routine threats experienced by developing plants are numerous. Compared to
animals, the sessile plant only requires the physical constituents of their
environments: light, ${\rm CO}_{2}$, ${\rm H}_{2}{\rm O}$ and minerals Gilroy
. A serious lack or imbalance of any of these can threaten viability as can a
substantial number of different mechanical, thermal, and biological
challenges. Phenotypic plasticity is a common behavioural response; tune the
phenotype to the experienced environmental conditions. Alterations occur
either via numbers of branch roots, shoot branches, leaves or buds, or by
changing their physical parameters, such as leaf area, stem thickness,
stiffness or proportions of different cell types. For instance, a lack of
water often results in increased root proliferation which penetrate to
increased soil depths. Plants are the basis of all food chains so predation is
common and often damaging. When predatory damage is recognised many plants
synthesise natural pesticides such as capsaicin, nicotine, or caffeine in
response. These kill the damaging insect or deter it from additional feeding.
It is estimated that as many as 100,000 such natural pesticides exist, the
product of an arms race between plants and predatory insects for 300 million
years Ames . Volatiles are also synthesised, and these attract predators of
the pest. In essence they function like a burglar alarm. Summing up, the
capability of efficiently processing information is vital to the survival of
plants.

The information-theoretic approach to understanding biological dynamics is
useful because it offers a way of modelling the dynamical behaviours of
biological systems and enables predictions. We believe it is particularly
advantageous in shedding light on the behaviour of green plants, for, it is
otherwise difficult to assess or measure informational quantities associated
with them. In particular, the mutual information involves the entropy of the
noisy observation $\xi$, which is certainly difficult to estimate from data.
But what do we learn from these information measures about plant biology? And
what are the open challenges from the informational perspective?

In view of the fact that plants form the basis of almost all food chains,
enhanced understandings of their behaviours, efficiencies, and capabilities
are becoming increasingly more important Gilroy . This information-based
approach will naturally contribute towards this endeavour. In particular, as
well as processing information, we expect that biological systems erase some
of the processed information. This is to be expected with transduction systems
using $[{\rm Ca}^{2+}]_{\rm i}$ since at the end of the first transient the
system is primed to respond to alternative signals. But erasure of information
inevitably requires energy consumption and heat production, in accordance with
Landauer’s principle in computation LR . In particular, the less efficient is
the computational capacity, the more energy it requires. Based on the recent
experiments with bean plants that can identify an object in its vicinity to
climb up Calvo , it has been conjectured that green plants must be able to
process information more efficiently than, say, the best computers available
DCB . Beyond this one estimate, little has been investigated or explored in
this direction. However, the information-based approach offers a new avenue of
research that can be supplemented by theoretical models Fields .

Determining the computational efficiency of green plants is an important
component in understanding how they can adapt to changing environments. The
observation made here that techniques of communication theory can be applied
to model biological dynamics (whether classical or quantum models are used),
even when there are no active agents transmitting the signal or purposefully
built communication channels, opens up the possibility of making a significant
progress in investigating and determining the informational efficiencies,
capabilities, and limits of plants. A deeper understanding of these issues may
even uncover ideas for technological advances. To make a significant progress
in this area, however, more data are required on plants. These include, in
particular, the understanding of how information is processed, how it is
stored, and how they are erased, in plants. Equally valuable information
concerns the number of information-storing units in plants (for processed
information, not genetic), and more generally how much energy is consumed by
plant cells. For instance, with a nanokelvin-resolution thermometer Reihani
it may be possible to measure heat production associated with erasure of
information in plants, which in turn will elucidate the computational
efficiencies of plants, or any other biological system for that matter. We
hope to explore these ideas elsewhere.

Acknowledgements. The authors thank Jim Al-Khalili, Bernhard Meister, and
Simon Saunders for stimulating discussions. DCB acknowledges support from the
John Templeton Foundation (grant 62210). The opinions expressed in this
publication are those of the authors and do not necessarily reflect the views
of the John Templeton Foundation.

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