# Quantitative Aspects, Engineering and Optimization of Bacterial Sensor
Systems

Florian Anderl, , Gabriela Salvadori, Mladen Veletic, Fernanda Cristina
Petersen and Ilangko Balasingham,  This work has been submitted to the IEEE
for possible publication. Copyright may be transferred without notice, after
which this version may no longer be accessible. Florian Anderl is with the
Department of Electronic Systems, Norwegian University of Science and
Technology, 7491 Trondheim, Norway. Gabriela Salvadori is with the
Intervention Centre, Oslo University Hospital,0424 Oslo, NorwayMladen Veletic
and Ilangko Balasingham are with the Intervention Centre, Oslo University
Hospital,0424 Oslo, Norway, and also with the Department of Electronic
Systems, Norwegian University of Science and Technology, 7491 Trondheim,
Norway.Fernanda Cristina Petersen is with the Institute of Oral Biology,
Faculty of Dentistry, University of Oslo, 0317 Oslo, Norway

###### Abstract

Bacterial sensor systems can be used for the detection and measurement of
molecular signal concentrations. The dynamics of the sensor directly depend on
the biological properties of the bacterial sensor cells; manipulation of these
features in the wet lab enables the engineering and optimization of the
bacterial sensor kinetics. This necessitates the development of biologically
meaningful computational models for bacterial sensors comprising a variety of
different molecular mechanisms, which further facilitates a systematic and
quantitative evaluation of optimization strategies. In this work, we dissect
the detection chain of bacterial sensors from a mathematical perspective from
which we derive, supported by wet-lab data, a complete computational model for
a _Streptococcus mutans_ -based bacterial sensor as a case example. We address
the engineering of bacterial sensors by investigating the impact of altered
bacterial cell properties on the sensor response characteristics, specifically
_sensor sensitivity_ and _response signal intensity_. This is achieved through
a sensitivity analysis targeting both the steady-state and transient sensor
response characteristics. Alongside the demonstration of suitability of our
methodological approach, our analysis shows that an increase of sensor
sensitivity, through a targeted manipulation of bacterial physiology, often
comes at the cost of generally diminished sensor response intensity.

###### Index Terms:

Bacterial Sensor, Molecular Communication, Biochemical Systems Theory,
Sensitivity Analysis

##  I Introduction

Molecular Communication (MC) is a biology-inspired communications paradigm
that is based on the notion of information encoded onto molecular carriers[1].
It is believed to possess vast potential to revolutionize the health care
field and much initial research is conducted to explore its capabilities[2].
MC is the basis for the envisioned _Internet of Bio-Nano Things_(IoBNT) in
which bio-nanomachines are connected in networks and exchange information over
an MC link [3]. IoBNT plays a crucial role in the future of theranostics in
form of implantable bio-nanosensors (BNS) continuously screening health-
relevant parameters and communicating this data in real-time over radio links
to a hub which could be connected to the Internet. A conceivable application
is, for example, the early-detection of bacterial or viral infections where
molecular markers of an ongoing infection are detected.

Much of research in MC either focuses on the theoretical implications of MC
systems and networks [4, 5, 6, 7] or quantitative information extraction from
biological systems [8, 9]. As MC ultimately aims at engineering and
prototyping practical solutions, selected works study feasible MC components.
The survey [10], for example, explores the requirements, communication-
theoretic properties and possible architectures for MC transmitter (Tx) and
receiver (Rx) components. It distinguishes between two main classes of
conceivable architectures: nanomaterial-based and biology-based designs. An
example for the former category is the graphene-based bioFET MC receiver
design [11]. In contrast, the latter category leverages synthetic biological
elements, e.g. , human or bacteria cells that can be genetically engineered
for a specific purpose. Here, bacteria are especially relevant due to the
relative ease of genetically engineering them for specific purposes which make
bacteria invaluable for various scientific, health-care related and industrial
applications.

Consequently, MC has acknowledged this potential in several works. Examples
include a concept where engineered bacteria are repurposed as transceivers
with different operational modes in an MC link [12]. Moreover, an MC
architecture with proton pumping bacteria was recently investigated [13], and
a bacterial receiver prototype using the rhamnose operon in a microfluidic
environment was presented alongside a computational model[14].

MC borrows the utility of bacteria as receivers from a key area of application
for synthetic bacteria: the _sensing_ of analytes. The deployment of bacteria
as biosensors for a variety of substances is a well-researched field in
microbiology and bio-engineering. Many works explore the principles and
techniques for engineering bacterial biosensors [15, 16, 17], and investigate
various areas of applications for bacterial biosensors, from environmental
monitoring [18, 19, 20] to medical applications [21, 22, 23, 24, 25].

While much of the related literature describes a more qualitative approach,
only few works attempt to derive quantitive measures for bacterial biosensors
[26]. Numerous computational models for dynamics arising from specific
bacterial mechanisms have been published [27, 28, 29, 30, 31, 32].
Nevertheless, to the knowledge of the authors none addresses the general
dynamics of bacterial systems with focus on its sensor properties. A lack of a
computational framework that describes the sensing dynamics of bacterial
sensors prevents us from proceeding with a systematic investigation of how the
dynamics can be altered to adhere to specific requirements of the engineered
bacterial sensor.

Most conceivable bacterial biosensors rely on the same key processes and
biochemical principles and thus have similar structures. This similarity is
useful for a computational analysis of these key processes and the resulting
qualitative and quantitative dynamics of bacterial sensors in general. This
analysis further allows for a systematic investigation of how these dynamics
can be altered to adhere to specific requirements on the bacterial sensor. In
this paper, we therefore conduct a computational analysis of bacterial sensors
and how bacteria can be engineered and optimized catering to the requirements
of building actual bacteria-based components for MC. We examine the general
components, integral features and critical processes of bacterial sensors with
an emphasis on sensing and detection. Through this, we summarize the
mechanisms that underlie general bacterial sensors and derive corresponding
mathematical models. Using methods from biochemical systems theory (BST) we
analyze the steady-state and transient dynamics of a _generic_ bacterial
sensor. We then proceed to derive expressions for the sensitivity and signal
intensity of the sensor. For these measures we then conduct a sensitivity
analysis and thereby identify critical parameters of the model and quantify
how changes in these affect the model dynamics. For example, similar to an
antenna in a telecommunications link or a radar dish, a bacteria-based MC
sensor- or receiver should exhibit a measure for gain.

As a case study, we present a concrete example of utilizing the proposed
general computational framework for building a computational model for
analyzing a bacterial sensor based on experimentation with the bacterial
species _Streptococcus mutans_. As basis for a reporter, we use the _luc_ gene
encoding the enzyme luciferase from the firefly _Photinus pyralisfirefly_
(further referrer to as _lucFF_), which is inserted into the genome of a
bacterial sensor strain. These modified bacteria produce the luciferase
enzyme, which upon addition of luciferin results in the emission of a light
signal proportional to a chemical target signal, functioning as sensors. This
allows, for instance, integration with a photodetector which further
facilitates integration with electronic elements.

The remainder of the paper is structured as follows. Sec. II discusses
important aspects of bacteria as biosensors and presents necessary biological
background information. Sec. III provides the theoretical framework which we
employ for our analysis. Sec. IV analyzes the discrete processes in a
bacterial sensor and summarizes mathematical modeling approaches for the
mechanisms. In Sec. V, we then present a mathematical model for a
_Streptococcus mutans-_ based sensing system and compare this model to our
empirical wet-lab data. Sec. VI presents a systematic scheme for analyzing a
bacterial sensor’s performance. Based on this, we comment on strategies for
optimization of bacterial sensors. Finally, Sec. VII concludes the paper and
outlines potential research building on this work.

##  II Bacteria as Biosensors

Figure 1:  Overview of general bacterial sensor mechanism. Molecular signal
$\mathsf{S}$, response regulator $\mathsf{RR}$, transcription factor
$\mathsf{TF}$ and reporter $\mathsf{P}$. (Illustration was created with
_BioRender.com_).

The general structure of a bacterial sensor is depicted in Fig. 1. The
bacteria’s molecular response pathway is activated by a target analyte or
effector eliciting the expression of reporter genes which, in the genome, are
inserted downstream of the regulated primary gene. The reporter proteins
resulting from this gene expression or rather their specific activity then are
a proxy for the analyte concentration. This mechanism is governed by several
distinct sub-processes emphasized in Fig. 1: _internalization, modification_
and _gene expression._

Internalization describes the uptake process of the extracellular signal into
the sensing cell either directly or by effect as is the case in, for example,
phosphorylation. In this case, the external signal binds to receptors on the
cell surface which elicits a response inside the cell. Subsequent to uptake,
the internalized analyte may undergo several modification steps involving a
_response regulator_(RR). Finally, the resulting _transcription factor_ (TF)
then regulates gene expression. These processes are detailed further below in
Sec. IV.

Various reporter mechanisms exist, most prominently: colorimetry, fluorescence
(FL) and bioluminescence (BL). Colorimetry is based on assay color changing
reactions elicited by, for example, beta-galactosidase resulting from the
expression of the _lacZ_ reporter gene. FL and BL rely on chemical reactions
producing light at specific wavelengths. FL resulting from expression of genes
like _gfp, egfp_ or _mCherry_ requires excitation from an external light
source. In contrast, BL functions without external activation which makes it
especially relevant for applications in which the bacterial sensor component
is integrated into an electronic device. The most important BL reporter
proteins are the bacterial luciferase variants LuxAB, LuxCDABE and firefly
luciferase LucFF. Various aspects influence the choice of reporter protein:
substrate availability (for example, BL relies on oxygen and luciferin),
sensitivity and, crucially, harvestability. A comprehensive overview of
reporters is given in [15].

We further distinguish between single-cell and population-based bacterial
sensors. The former exhibits a cellular granularity since the response of
single cells is considered [33, 34]. In contrast, in the population-based
approach the response of the entire population of the sensing strain is
measured. The stochasticity inherent in molecular mechanisms (especially
relevant at low analyte concentrations) and gene expression results in
generally non-deterministic single-cell bacterial sensor dynamics. This
implies heterogenous reporter expression levels across a sensor population.
This heterogeneity can be measured through microscopy imaging and be
advantageous for certain sensing applications as it can potentially resolve
spatial information.

In this paper, we restrict our treatment to an analysis of whole-cell
bacterial sensors on population-basis deployed in an aqueous assay. This
implies a homogeneous concentration of a target analyte in the assay and each
viable bacterial cell being exposed to the same concentration of target
analyte. We assume further that each viable bacterial cell contributes equally
to the overall sensor response, i.e., we ignore the natural heterogeneity of
gene expression across a bacterial population. We justify this by our aiming
at deriving a model for the _mean bacterial cell response_ ,__ as discussed in
Sec. V-C.

We further distinguish between two general sensing scenarios:

##### Scenario 1 - Abundance of Analyte

In this case, the analyte concentration in the sensing cavity is sufficiently
high so that it is not influenced or noticeably diminished by the sensor
dynamics itself. This could be the case if there exists a constant supply of
analyte into the sensing cavity which is a reasonable assumption for cases of
e.g., acute infection with a virulent bacterial strain that is in the
stationary phase of its growth cycle. If the virulent strain emits a specific
detectable signal (e.g., through _Quorum Sensing_), we can assume an abundant
signal supply. In this case, the target analyte enters the computational model
describing the sensor as _constant_ _independent variable_. As a result, a
bacterial sensor cell reaches a non-zero steady-state response level.

##### Scenario 2 - Scarcity of Analyte

When the effector signal is at a low concentration and depleted without
experiencing a positive influx it must be included in the computational model
as an additional equation as the bacterial sensor _itself_ affects the
concentration of analyte.

In Sec. VI, we use _Scenario 1_ and _Scenario 2_ for a systematic steady-state
and dynamic optimization analysis, respectively.

##  III Theoretical Framework

Bacterial sensors can be analyzed as biochemical systems at different levels
of abstraction. Low concentrations of involved biochemical species mandate
stochastic and particle-based models accounting for the probabilistic nature
of molecular interactions. Monte-Carlo simulations allow for the synthesis of
deterministic trajectories from the stochastic models. Alternatively, direct
deterministic models based on reaction-rate equations (RREs) are often a
simpler approach while retaining the capability of capturing the system
dynamics accurately.

In this section, we give a brief overview of BST as it provides methods that
are crucial for our analysis of a bacterial sensor. In the following section,
we then outline dynamics of various bacterial sensors and subsequently apply
the BST methodology to the derived models of bacterial sensors to provide a
systematic analysis of their performance, thus revealing potential starting
points for their optimization.

###  III-A Biochemical Systems Theory

BST studies methods for mathematical modeling of biochemical systems [35]. It
does so by deriving simplistic approximative models that allow for facilitated
mathematical analysis while retaining the dynamics observed in the data.__ The
analysis of a biochemical system generally begins with the identification of
structural relationships in the studied system which can be comprehensively
illustrated by graphical representations (as done in Sec. IV, cf. Fig. 2). The
dynamics of a system with various constituent species $X_{1},X_{2},...,X_{n}$
are generally described by a set of ordinary-differential-equations (ODEs)
parametrized by $\mathbf{\Theta}_{R}=\\{\theta_{1},...,\theta_{F}\\}$ as

$\displaystyle\dot{X}_{1}$
$\displaystyle=v_{1}^{+}(X_{1},...,X_{n},...,X_{n+m};\mathbf{\Theta}_{R})$
$\displaystyle\vdots$ $\displaystyle\hskip 28.45274pt-
v_{1}^{-}(X_{1},...,X_{n},...,X_{n+m};\mathbf{\Theta}_{R})$
$\displaystyle\dot{X}_{n}$
$\displaystyle=v_{n}^{+}(X_{1},...,X_{n},...,X_{n+m};\mathbf{\Theta}_{R})$
$\displaystyle\hskip 28.45274pt-
v_{n}^{-}(X_{1},...,X_{n},...,X_{n+m};\mathbf{\Theta}_{R})$ (1)

where the flux terms $v_{i}^{+/-},i=1,...,n$ are either positive fluxes adding
to a pool of species $X_{i}$ or negative when depleting it. These fluxes are
in general functions of the system’s $n$ internal state variables
$\mathbf{X}_{\mathsf{int}}=\left[X_{1},...,X_{n}\right]^{T}$ and $m$ external
variables $\mathbf{X}_{\mathsf{ext}}=\left[X_{n+1},...,X_{n+m}\right]^{T}$.
The fluxes specifically depend on a subset of variables identified in the
structural analysis of the biochemical system under investigation. The flux
terms can assume various different functional forms: from basic RREs to terms
describing complex mechanisms as, for example, _Michaelis-Menten-kinetics_ or
_Competitive Inhibition_.

It is mathematically convenient to employ an universal form for representing
the fluxes $v_{i}^{+/-}$ regardless of the dynamics governing the specific
case and this form is provided by the _Power-Law-Formalism._ As in a classical
mass-action derived RRE, a single flux is modeled as a product of a kinetic
rate constant $\gamma$ and involved species among which each are raised to a
power of a specific kinetic rate (or kinetic orders) $f$, i.e.,

$v_{i}^{+/-}=\gamma_{i}\prod_{j=0}^{n+m}X_{j}^{f_{ij}},\quad i=1,...,n.$ (2)

In (1), a flux $v_{i}^{+/-}$ can consist of several distinct processes
(chemical reactions, molecular interactions, etc.), i.e.,
$v_{i}^{+/-}=\sum_{k=1}^{T_{i}}v_{ki}^{+/-}$. The choice of representing a
biochemical system with aggregated flux terms $v_{i}^{+/-}$or, alternatively,
distinguishing single process rates $v_{ki}^{+/-},k=1,...,T_{i}$ yields either
an _S-System_ or a _General-Mass-Action_(GMA) system, respectively.111For the
sake of completeness, a GMA system is given by
$\dot{X}_{i}=\sum_{k=1}^{T_{i}}\pm\gamma_{ik}\prod_{j=1}^{n+m}X_{j}f^{ikj},\quad
i=1,...,n$ All positive (negative) terms in the above equation are aggregated
into the first (second) term on the right hand side of (3). As the name
implies, GMA systems are generalized mass action __ formulations and allow for
non-integer valued kinetic orders. In that sense, allowing the kinetic orders
to assume any real number renders a GMA model more flexible. Consequently, the
canonical BST formulation of an S-system with $n$ dependent and $m$
independent variables has the format [36]

$\dot{X}_{i}=\alpha_{i}\prod_{j=1}^{n+m}X_{j}^{g^{ij}}-\beta_{i}\prod_{j=1}^{n+m}X_{j}^{h^{ij}},\quad
i=1,...,n.$ (3)

Rate terms of the form (2) are in general _approximations_ to the real
dynamics of the specific process. Consequently, the parameters
$\mathbf{\Theta}_{S}=\left\\{g_{ij},h_{ij},\alpha_{i},\beta_{i}\right\\},i=1,...,n;j=1,...n+m$
of an S-system can be derived from the base system as [36]

$\displaystyle g_{ij},h_{ij}$
$\displaystyle=\left.\frac{\partial\log\left(\sum_{k=1}^{T_{i}}v_{k}^{+,-}\right)}{\partial\log
X_{j}}\right|_{\mathbf{X=X}_{0}}$
$\displaystyle=\left.\frac{\partial\left(\sum_{k=1}^{T_{i}}v_{k}^{+,-}\right)}{\partial
X_{j}}\right|_{\mathbf{X=X}_{0}}\frac{X_{j}}{\left(\sum_{k=1}^{T_{i}}v_{k}^{+,-}\right)},$
(4) $\displaystyle\alpha_{i},\beta_{i}$
$\displaystyle=v_{i}^{+,-}\left(\mathbf{X}_{0}\right)\prod_{j=1}^{n+m}X_{j}^{-g_{ij},h_{ij}}.$
(5)

In (4) and (5) $\mathbf{X}_{0}=\left[X_{1_{0}},...,X_{n_{0}}\right]^{T}$ is
the _operating point_ at which the S-system is constructed. At this operating
point, equality holds between the base biochemical system and its
corresponding S-system form.

We introduce S-systems here since, in Sec. VI, we make use of their convenient
mathematical properties which facilitate analytic steady-state analysis. We
briefly outline these mathematical properties in the following.

###  III-B Steady-State Analysis

The canonical S-system form has convenient mathematical properties which allow
for the direct analytic calculation of its steady-state solution in the log-
space as [36]

$\mathbf{y}_{\mathrm{int}}=\mathbf{L}\mathbf{y}_{\mathrm{ext}}+\underset{\mathbf{m}}{\underbrace{\mathbf{M}\mathbf{b}}}$
(6)

where the matrix $\mathbf{L}\in\mathbb{R}^{n\times m}$ contains the _log-
gains_ which relate changes in logarithms of independent (external) variables
in $\mathbf{y}_{\mathsf{ext}}\in\mathbb{R}^{m\times 1}$ to changes in
logarithm of dependent (internal) variables
$\mathbf{y}_{\mathsf{int}}\in\mathbb{R}^{n\times 1}$.

Entries of $\mathbf{L}$, $\mathbf{M}$ and $\mathbf{b}$ are composed of
functions of the S-system parameters $\mathbf{\Theta}_{S}$ and are thus
functions of $\mathbf{\Theta}_{R}$, i.e., the parameters of the original RRE
following (4) and (5). Eq. (6) implies that the independent variables in
$\mathbf{y}_{\mathrm{int}}$ are _linear_ functions of the independent
variables in $\mathbf{y}_{\mathrm{ext}}$ with the slope of the resulting line
given by the entries $l_{ij}=\frac{\partial X_{i}}{\partial
X_{j}}\frac{X_{j}}{X_{i}},i=1..,n;j=1,...,m$ of $\mathbf{L}$ and the constant
offset given by element of the vector $\mathbf{m}$ resulting from the matrix-
vector product $\mathbf{M}\mathbf{b}$.

We use the form of (6) in the application to a bacterial sensor model in Sec.
VI to directly obtain steady-state figures that can be linked to general
sensors properties.

Considering the optimization and engineering of bacterial sensors as the main
goal of this work, we then proceed with an analysis of how these sensor
properties depend on the sensor model parameters. Formally, we achieve that
through a _sensitivity analysis._

###  III-C Steady-State Sensitivities

From (6) one can directly obtain the sensitivities of the steady-state
solutions $\mathbf{y}_{i}$ with respect to the S-system parameters
$\mathbf{\Theta}_{S}$ and in extension of $\mathbf{\Theta}_{R}$. The
_relative_ sensitivities $S_{l_{ij},\theta_{k}},S_{m_{i},\theta_{k}}$ are
calculated for $l_{ij}\in\mathbf{L}$ and $m_{i}\in\mathbf{m}$ using the chain-
rule as [37]

$\displaystyle S_{l_{ij},\theta_{k}}$ $\displaystyle=\frac{\partial
l_{ij}}{\partial\theta_{k}}\left|\frac{\theta_{k}}{l_{ij}}\right|=\sum_{f\in\tilde{\Theta}_{S}}\frac{\partial
l_{ij}}{\partial f}\frac{\partial
f}{\partial\theta_{k}}\times\left|\frac{\theta_{k}}{l_{ij}}\right|,$ (7)
$\displaystyle S_{m_{i},\theta_{k}}$ $\displaystyle=\frac{\partial
m_{j}}{\partial\theta_{k}}\left|\frac{\theta_{k}}{m_{j}}\right|=\sum_{f\in\Theta_{S}}\frac{\partial
m_{i}}{\partial f}\frac{\partial
f}{\partial\theta_{k}}\times\left|\frac{\theta_{k}}{m_{i}}\right|,$ (8)
$\displaystyle\quad\theta_{k}\in\mathbf{\Theta}_{R},\tilde{\mathbf{\Theta}}_{S}\subseteq\mathbf{\Theta}_{S},i=1,...n,j=1,...,m.$

Eqs. (7) and (8) evaluate the _percentage_ changes in $\mathbf{L}$ and
$\mathbf{m}$ due to infinitesimal percentage changes in $\theta_{k}\in$
$\mathbf{\Theta}_{R}$. We use these measures to investigate the contribution
of biological parameters to the steady-state behavior of bacterial sensors.

###  III-D Dynamics Sensitivities

In addition to the steady-state treatment, we conduct a _dynamic sensitivity
analysis_ for bacterial sensor models. Here we assume that the investigated
biochemical system has _no_ independent variables as the kinetics of the
analyte state variable are now affected by the system dynamics.

With $\mathbf{f}_{c}\in\mathbb{R}^{n\times 1}$ comprising elements
$\partial\dot{X_{i}}/\partial\theta_{k},i=1,...,n,\theta_{k}\in\mathbf{\Theta}_{R}$,
$\mathbf{J}\in\mathbb{R}^{n\times n}$ as the Jacobian with entries
$\partial\dot{X}_{i}/\partial X_{j},i,j=1,...,n$ and dynamic sensitivity
vector $\mathbf{z}_{k}\in\mathbb{R}^{n\times 1}$ with entries $\partial
X_{i}/\partial\theta_{k}$[38],

$\dot{\mathbf{z}}_{k}=\mathbf{f}_{c}+\mathbf{J}\mathbf{z}_{k},\quad k=1,...,F$
(9)

denote the differential equations of the dynamic sensitivities with respect to
parameter $\theta_{k}$. An ODE system consisting of $(F\cdot n)$ equations of
the form (9) can then be solved numerically to yield the final dynamic
sensitivities $\mathbf{z}$ that describe the temporal evolution of the
sensitivities. This provides insight into how different phases of the time-
course dynamics of a variable $X_{i}$ responds to changes in parameters
$\mathbf{\Theta}_{R}$.

One can further include the initial conditions of $\mathbf{X}$ into
$\mathbf{\Theta}_{R}$ to investigate their effects on the system dynamics. In
fact, by calculating the _relative dynamic sensitivity_ with respect to the
initial condition $X_{j}(0)$ of the _dependent_ variable __ $X_{j}$, i.e.,

$S_{i,X_{j}(0)}=\frac{\partial X_{i}}{\partial
X_{j}(0)}\frac{X_{j}(0)}{X_{i}},\quad i,j=1,...,n$ (10)

a measure is obtained which is very similar to the _dynamic log-gain_ [39].__
The dynamic log-gain can be used to evaluate the influence of changes in
independent variables on the time-course of the investigated state-variable.
Eq. (10), in contrast, assumes $X_{j}$ to be dependent, hence the sensitivity
with respect to its initial condition is measured.

We can then proceed to calculate the sensitivities of $S_{i,X_{j}(0)}$ with
respect to other model parameters $\theta_{k}\in\mathbf{\Theta}_{R}$ by
solving the following differential equation (for derivation see Appendix B):

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial
S_{i,X_{j}(0)}}{\partial\theta_{k}}\right)$
$\displaystyle=-\left(\frac{\partial
S_{i,X_{j}(0)}}{\partial\theta_{k}}X_{i}+S_{i,X_{j}(0)}\frac{\partial
X_{i}}{\partial\theta_{k}}\right)\frac{\dot{X_{i}}}{X_{i}^{2}}$
$\displaystyle+X_{j}(0)\frac{\partial X_{i}}{\partial
X_{j}(0)}\left(2X_{i}^{-3}\dot{X_{i}}-X_{i}^{-2}\frac{\partial\dot{X_{i}}}{\partial\theta_{k}}\right).$
(11)

Eq. (11) constitutes the dynamic sensitivity analysis equivalent of (7). As it
shares terms with (9), it can be integrated with $\dot{\mathbf{z}}_{k}$ into
an extended ODE system.

##  IV Dissecting the Bacterial Sensor Detection Chain

Figure 2: Structural diagram of a bacterial sensor. The dotted arrows indicate
that all species are subject to degradation.

The signal transduction pathway depicted in Fig. 2 shall serve as reference
for the analysis of the various distinct steps in the detection chain of a
bacterial sensor and the respective computational models. In the most general
sense, the target analyte $\textsf{S}_{\textsf{ext}}$ is _internalized_ by a
sensor cell and then potentially undergoes _modification_ in interaction with
response regulator RR. The resulting transcription factor TF subsequently
drives the _expression_ of the gene coding for the reporter protein P.
Importantly, the internalization of the effector $\textsf{S}_{\textsf{ext}}$
can either occur directly or indirectly by effect only. This is the case of
the _phosphorylation_ of an internal response regulator caused by an effector-
bound cell surface receptor. We dissect the detection chain into its distinct
processes and analyze their mathematical models. These models are initially
provided as ordinary chemical reaction rate equations, since models of this
form are most intuitive and generally a good starting point for a
computational analysis. We note that parts of this section do not present new
findings but collects information from various sources. We deem it necessary
to proceed in this fashion as the presented information is crucial to
comprehensively discussing bacterial sensor dynamics.

###  IV-A Internalization

The first step in the bacterial sensor detection chain is the uptake of the
target analyte $\mathsf{S_{ext}}$ into the bacterial sensor cells (Fig. 2).
The variety of bacteria strains exhibit a multitude of different uptake
mechanisms that vary in their biochemical dynamics. The transport mechanisms
can be categorized into the following groups:

#### Diffusion

In many bacteria nutrients or molecular signals _diffuse_ into the bacterial
cells across the cell wall. This is a passive process and requires no energy
expenditure by the cell. This process can generally be described by Fick’s law
of diffusion across a membrane as [40, 29]

$\displaystyle\frac{\mathrm{d}\mathsf{S}_{\mathsf{int}}}{\mathrm{d}t}$
$\displaystyle=-q_{D}\left(\frac{\mathsf{S}_{\mathsf{int}}}{V_{\mathrm{i}}}-\frac{\mathsf{S}_{\mathsf{ext}}}{V_{\mathrm{e}}}\right)-\delta_{\mathsf{s}}\mathsf{S_{int}}$
(12)

where $\mathsf{S_{ext}}$ and $\mathsf{S_{int}}$ denote the external and
internal amount of target analyte, respectively, $V_{\mathrm{i}}$ and
$V_{\mathrm{e}}$ are the intracellular and extracellular volumes and
$\delta_{\mathsf{s}}$ is the degradation rate constant of $\mathsf{S_{int}}$.
The constant $q_{D}$ collects the cell wall thickness, the diffusion
coefficient and the cell membrane surface area. The expression in parentheses
models the concentration gradient across the cell wall. If the external
analyte amount $\mathsf{S_{ext}}$ is sufficiently high or subject to a
constant supply, corresponding to _Scenario 1_ described in Sec. V, it can be
treated as constant $\mathsf{\widetilde{S}_{ext}}$.

Considering that the uptake process is linear and time-invariant (LTI), it can
thus be represented in form of an impulse response given by

$h_{\mathsf{up}}(t)=\mathcal{F}^{-1}\left\\{\frac{\frac{q_{D}}{V_{\mathrm{e}}}}{j\omega+\frac{q_{D}}{V_{\mathrm{i}}}+\delta_{s}}\right\\}=u(t)\frac{q_{D}}{V_{\mathrm{e}}}e^{-\left(\frac{q_{D}}{V_{\mathrm{i}}}+\delta_{s}\right)t}$
(13)

where $u(t)$ is the unit step function and $\mathcal{F}^{-1}\\{\cdot\\}$
denotes the inverse Fourier transform.

#### Facilitated Diffusion and Active Transport Mechanisms

In facilitated diffusion molecules are transported into the cell by means of
membrane proteins or permeases. Facilitated diffusion is a passive process and
is driven by a concentration gradient. Active transport mechanisms generally
require an energy expenditure. As the number of transport proteins on the cell
surface is finite, both processes are inherently saturable regardless of the
external concentration which is the main difference to free diffusion.

With the signal or effector molecule $\mathsf{S_{ext}}$ and the surface
permease $\mathsf{M}$, the reaction equation for this process is given by

$\ce{\mathsf{S_{ext}}+\mathsf{M}<=>[k_{\mathrm{b}}][k_{-\mathrm{b}}][\mathsf{S_{ext}}\mathsf{M}]->[k_{\mathrm{int}}]\mathsf{M}+\mathsf{S_{int}}}$
(14)

where $k_{\mathrm{b}}$ ($k_{-\mathrm{b}}$) is the forward (backward) reaction
rate constant of the binding reaction and $k_{\mathrm{int}}$ is the
internalization rate constant.

Mathematically, the molecular uptake mechanisms follow saturation dynamics and
can thus be described using Michaelis-Menten kinetics as

$\displaystyle\frac{\mathrm{d}\mathsf{S_{int}}}{\mathrm{d}t}^{+}$
$\displaystyle=\kappa_{\mathrm{up}}\left(\frac{\mathsf{S_{ext}}}{K_{\mathrm{up}}+\mathsf{S_{ext}}}\right)$
(15)

where the parameters $\kappa_{\mathrm{up}}$ and $K_{\mathrm{up}}$ are the
maximum internalization rate and concentration at half maximum, respectively.
They are explicitly derived from (14) as
$\kappa_{\mathrm{up}}=\mathsf{M}_{0}k_{\mathrm{int}}$, where $\mathsf{M}_{0}$
is the initial permease amount at $t=0$ and
$K_{\mathrm{up}}=\frac{k_{-\mathrm{b}}+k_{\mathrm{int}}}{k_{\mathrm{b}}}$,
following Michaelis-Menten derivations. Similar modeling approaches have been
discussed in the literature [41, 42]. For instance, in the literature
$K_{\mathrm{up}}$ is occasionally used to characterize the kinetic uptake
properties of a specific permease [43]. Eq. (15) is non-linear and it is thus
not straightforward to derive a transfer function or impulse response for it.
However, for relatively low concentrations $\mathsf{S_{ext}}\ll
K_{\mathrm{up}}$, eq. (15) can be treated as linear which facilitates
derivation of the corresponding LTI model which is given by

$h_{\mathsf{up}}(t)=u(t)\frac{\kappa_{\mathrm{up}}}{K_{\mathrm{up}}}e^{-\delta_{s}t}.$
(16)

###  IV-B Transcription Factor Activation and Analyte Modification

Inside the bacterial sensor cell, the effector molecules either directly act
as transcription factor $\mathsf{TF}$ for a specific gene or undergo a series
of reactions and interactions with regulatory proteins or response regulators
$\mathsf{RR}$, to form the $\mathsf{TF}$ complexes. This is described by the
reaction equation

$\displaystyle\ce{a\mathsf{S_{int}}+b\mathsf{RR}}$
$\displaystyle\ce{<=>\underset{\mathsf{TF}}{\underbrace{[\mathsf{S_{int}}^{a}\mathsf{RR}^{b}]}}}.$
(17)

$a$ and $b$ denote the moieties of the species partaking in reaction (17).
Depending on the precise reaction nature, the conceivable reaction dynamics
range from linear (in case of a monomer transcription factor) to highly non-
linear (for multimers). This is reflected by the reaction rate equations

$\displaystyle\frac{\mathrm{d}\mathsf{TF}}{\mathrm{d}t}$
$\displaystyle=k_{1}\left(\frac{\mathsf{S_{int}}}{V_{\mathrm{i}}}\right)^{a}\mathsf{RR}^{b}-\delta_{\mathsf{TF}}\mathsf{TF}$
(18)

with $\delta_{\mathsf{TF}}$ denoting the degradation rate constant of
$\mathsf{TF}$. In (18), the division of $\mathsf{S_{int}}$ by the bacterial
cell volume $V_{\mathrm{i}}$ is due to the units of the bi-molecular rate
constant $k_{1}\left(${\mathrm{\SIUnitSymbolMicro
m}}^{3a}\text{\,}{\mathrm{min}}^{-1}$\right)$ [31].

Due to the product of two state variables $\mathsf{S_{int}}$ and $\mathsf{RR}$
in (18), and the resulting non-linearity, the derivation of a system function
or impulse response is challenging. With the assumptions of singular moieties
on part of $\mathsf{S_{int}}$, i.e., $a=1$ and of abundance of response
regulator with $\mathsf{RR}\gg\mathsf{S_{int}}$, it is, however, feasible. In
this case $\mathsf{RR}^{b}$ can be treated as a constant parameter because the
reaction with $\mathsf{S_{int}}$ does not substantially change the
concentration of $\mathsf{RR}^{b}$. Then, the impulse response can be given by

$h_{\mathsf{mod}}(t)=\mathcal{F}^{-1}\left\\{\frac{\frac{k_{1}}{V_{\mathrm{i}}}\mathsf{RR}^{b}}{j\omega+\delta_{\mathsf{TF}}}\right\\}=u(t)\frac{k_{1}}{V_{\mathrm{i}}}\mathsf{RR}^{b}\text{e}^{-\delta_{\mathsf{T}}t}.$
(19)

In case of a multimerization with $a>1$, the process is highly non-linear and
an impulse response can only be derived by employing more elaborate techniques
such as Volterra series analysis [8].

As stated above, a signal transduction pathway can comprise an indirect uptake
mechanism where the analyte molecule is not entering the cell directly, but
binds to a surface permease leading to the modification of the intracellular
response regulator. This process is termed _phosphorylation._

####  IV-B1 Phosphorylation

A common mechanism by which intracellular $\mathsf{RR}$ is turned on is
_phosphorylation_. When the target analyte extracellularly binds to the
receptor on the cell surface, a phosphate group is transferred from the
intracellular surface receptor component to the free $\mathsf{RR}$. For an
arbitrary $\mathsf{RR}$ (with phosphorylated form $\mathsf{RR}_{\mathrm{p}}$
which then acts as transcription factor) and a surface receptor $\mathsf{C}$
(with bound state $\mathsf{C_{b}}$) this can be modeled as [31]

$\displaystyle\frac{\mathrm{d}\mathsf{C_{b}}}{\mathrm{d}t}$
$\displaystyle=k_{\mathrm{B}}\frac{\mathsf{S_{ext}}}{V_{\mathrm{e}}}\mathsf{C}-k_{\mathrm{B}_{-}}\mathsf{C_{b}}-\delta_{\mathsf{C_{\mathrm{b}}}}\mathsf{C_{b}}$
(20) $\displaystyle\frac{\mathrm{d}\mathsf{RR}_{\mathrm{p}}}{\mathrm{d}t}$
$\displaystyle=k_{\mathrm{p}}\mathsf{RR}\mathsf{\frac{C_{b}}{V_{\mathrm{i}}}}-k_{\mathrm{p}_{-}}\mathsf{RR}_{\mathrm{p}}-\delta_{\mathsf{RR}_{\mathrm{p}}}\mathsf{RR}_{\mathrm{p}}.$
(21)

$k_{\mathrm{B}}$, $k_{\mathrm{B}_{-}}$, $k_{\mathrm{p}}$ and
$k_{\mathrm{p}_{-}}$ are the reaction rate constants for binding, unbinding,
phosphorylation and dephosphorylation, respectively. We note that the total
number of surface receptors in free or bound state is constant, i.e.,
$\widetilde{\mathsf{c}}=\mathsf{C}+\mathsf{C_{b}}$ following the _conservation
of mass_ rule. Under the condition that the phosphorylation of $\mathsf{RR}$
does not affect its stability
($\delta_{\mathsf{RR}}=\delta_{\mathsf{RR}_{\mathrm{p}}}$), it also applies
that $\widetilde{\mathsf{RR}}=\mathsf{RR}+\mathsf{RR}_{\mathrm{p}}$. From this
follows that (20) and (21) can be written without explicit dependence on
$\mathsf{C}$ and $\mathsf{RR}$ as

$\displaystyle\frac{\mathrm{d}\mathsf{C_{b}}}{\mathrm{d}t}$
$\displaystyle=k_{\mathrm{B}}\mathsf{\frac{\mathsf{S_{ext}}}{V_{\mathrm{e}}}}\left(\widetilde{\mathsf{c}}-\mathsf{C_{b}}\right)-k_{\mathrm{B}_{-}}\mathsf{C_{b}}-\delta_{\mathsf{C_{\mathrm{b}}}}\mathsf{C_{b}}$
(22)
$\displaystyle\frac{\mathrm{d}\mathsf{\mathsf{RR}_{\mathrm{p}}}}{\mathrm{d}t}$
$\displaystyle=k_{\mathrm{p}}\mathsf{\frac{C_{b}}{V_{\mathrm{i}}}}\left(\widetilde{\mathsf{RR}}-\mathsf{\mathsf{RR}_{\mathrm{p}}}\right)-k_{\mathrm{p}_{-}}\mathsf{\mathsf{RR}_{\mathrm{p}}}-\delta_{\mathsf{RR}_{\mathrm{p}}}\mathsf{\mathsf{\mathsf{RR}_{\mathrm{p}}}}.$
(23)

###  IV-C Transcription and Translation

In the final step of the detection chain of a bacterial sensor, the
$\mathsf{TF}$ complex binds to DNA segments known as promoter and thereby
regulates gene expression. This regulation can manifest either as up-
regulation (activation) or repression (inhibition; not treated here) of gene
expression. The precise mechanism that underlies this essential process
justifies closer inspection which is provided in Appendix Sec. A. This
analysis finally yields the reaction rate equation for a reporter protein
$\mathsf{P}$ as

$\frac{\mathrm{d}\mathsf{P}}{\mathrm{d}t}=\alpha_{\mathrm{T}}\left[\kappa_{\mathrm{B}}+\kappa_{\mathsf{P}}\left(\frac{\mathsf{TF}^{n_{\mathrm{Hill}}}}{K^{n_{\mathrm{Hill}}}+\mathsf{TF}^{n_{\mathrm{Hill}}}}\right)\right]-\delta_{\mathsf{P}}\mathsf{P}.$
(24)

$\alpha_{\mathrm{T}}$ denotes the translation efficiency,
$\kappa_{\mathrm{B}}$ is the basal transcription rate, $\kappa_{\mathsf{P}}$
is the maximum up-regulated transcription rate, $K$ is the Michaelis-Menten
constant for transcription, $n_{\mathrm{Hill}}$ is the Hill coefficient and
$\delta_{\textsf{P}}$ the decay rate constant of the reporter protein.

With $\mathsf{TF}\ll K$, $n_{\mathrm{Hill}}=1$ and assuming no basal
transcription ($\kappa_{\mathrm{B}}=0$), eq. (24) can be modeled as an LTI
system, i.e.,

$h_{\mathsf{T}}(t)=u(t)\alpha_{\mathrm{T}}\frac{\kappa_{\mathsf{P}}}{K}e^{-\delta_{\mathsf{P}}t}.$
(25)

##  V Bacterial Sensor Models

With the mathematical descriptions for the single steps in the detection
chain, it is now possible to construct full mathematical models of different
bacterial sensor architectures. First, we outline under which conditions a
bacterial sensor follows an LTI model. We then briefly discuss how the effects
of bacterial population dynamics can be accounted for in a population-level
bacterial sensor. Finally, we present a full bacterial sensor model of a real
bacterial sensing system based on the strain _Streptococcus mutans._

###  V-A Linear Model for Bacterial Sensor

Figure 3: Block diagram of the LTI model of a bacterial sensor.

It is possible to construct a _linear_ bacterial sensor model if the following
conditions are met:

  * •

Uptake via diffusion _or_ active uptake modeled using Michaelis-Menten
kinetics with analyte concentrations $\mathsf{S_{ext}}\ll K_{\mathrm{up}}$

  * •

Analyte modification following (19) with analyte moiety of $a=1$

  * •

Transcription and translation following a Michaelis-Menten model with
$n_{\mathrm{Hill}}=1$ with $\mathsf{TF}\ll K$

The full detection chain of the sensor is then described by an LTI model
visualized in Fig. 3 where each step in the detection chain, i.e., uptake,
modification and transcription/translation is symbolized by a block
representing the process’s impulse response. The significance of this approach
is that the output protein P can be expressed directly as

$\mathsf{P}(t)=\underset{h_{\mathsf{BS}}(t)}{\underbrace{h_{\mathsf{up}}(t)*h_{\mathsf{mod}}(t,\mathsf{RR})*h_{\mathsf{T}}(t)}}*\mathsf{S_{ext}}(t)$
(26)

where $*$ denotes convolution and $h_{\mathsf{BS}}(t)$ the total impulse
response of the bacterial sensor cell. In (26), the parametrization of
$h_{\mathsf{mod}}(t,\mathsf{RR})$ by $\mathsf{RR}$ is made explicit to
emphasize the condition under which (19) applies.

###  V-B Whole Cell Bacterial Sensor - Population Dynamics

A population-level bacterial sensor harvests the bioluminescent response of an
entire bacterial population; it is thus necessary to incorporate the
population size into the model. Consequently, $\nu(t)$ models the size of the
bacterial sensor strain __ at time $t$.

Bacterial populations undergo specific phases in the course of their growth
cycle: the lag phase, the log-phase, the stationary phase and the death phase.
Numerous mathematical models and iterations on the classical sigmoidal growth
model for describing these dynamics have been published [44]. Our model is
based on a variation of the original Verhulst growth model [45], where our
contribution is the incorporation of a _delay_ into the the governing
differential equation.

The sensor strain’s growth dynamics are then more accurately described by

$\frac{\mathrm{d}\nu}{\mathrm{d}t}=\mu\nu(t)\left(1-\frac{\nu(t-\tau)}{K_{\nu}}\right)^{b_{\nu}}$
(27)

where $\mu$ denotes the population growth constant and the carrying capacity
$K_{\nu}$ reflects the limits a bacterial population’s growth is subjected to
in a specific environment. $b_{\nu}$ is an additional scaling parameter that
describes the populations capability to exceed the carrying capacity
($b_{\nu}>1$) or to stay below it ($b_{\nu}<1$). Eq. (27) is a delay-
differential equation with delay $\tau$ and is chosen to capture an overshoot
in the populations growth cycle where its size initially exceeds the systems
carrying capacity and subsequently levels off. We chose this modeling approach
to explain our experimental data obtained from wet-lab experiments (Fig. 4).

Figure 4:  Experimental bacterial optical density (OD600) for _Streptococcus
mutans_ vs. delay-differential equation model with fitted parameters
$K_{\nu}=7.5603\times 10^{8}$,
$\mu=$0.009\,022\,4\text{\,}{\mathrm{min}}^{-1}$$,
$\tau=$80.5438\text{\,}\mathrm{min}$$ and $b_{\nu}=1$.

###  V-C Streptococcus mutans as Sensor for XIP

For the bacterial sensor investigated in this paper, we choose the bacterium
_Streptococcus mutans_(_S. mutans_)_._ This Gram-positive bacterial strain is
found in the human oral cavity and is the primary cause for tooth decay [46].
In _S. mutans_ , natural competence, i.e., the ability of bacteria to take up
naked DNA from their environment and incorporate it into their genomes as
survival- and adaption strategy, is regulated by pheromones/peptides [47]. _S.
mutans_ exhibits two signaling pathways over which two distinct peptides
function as signals: the CSP and the XIP pathway. Both pathways activate the
promoter of _sigX,_ the master regulator of competence.

In this paper, we focus on the distinct XIP signaling pathway. Extracellular
XIP (SigX-inducing peptide) is imported by the cell-membrane bound
oligopeptide permease ($\mathsf{OPP}$). The imported XIP activates the free
form of the intracellular response regulator ($\mathsf{\mathsf{ComR_{f}}})$ by
building a dimer (i.e., involving two molecules of XIP and free ComR each)
$\mathsf{ComR_{m}}$ which, in turn, activates the transcription of _sigX._
Recent studies suggest that $\mathsf{ComR_{m}}$ further acts as transcription
factor for _comR_[48] which implies a positive feedback loop in the XIP
signaling pathway (Fig. 5). In a wild-type _S. mutans_ strain,
$\mathsf{ComR_{m}}$ further acts as a transcription factor for _comS_ which
codes for the intracellular XIP-precursor ComS. Since _S. mutans_ produces and
secretes XIP (or ComS) naturally, we knock out the _comS_ gene in the _S.
mutans_ sensor strain to ensure that no endogenous XIP is produced thereby
eliminating this source of ‘noise’.

Figure 5: Transcriptional activation of _sigX/lucFF_ operon in engineered _S.
mutans_ strain _._(Illustration was created with _BioRender.com_).

For this experiment we use the _lucFF_ gene as a reporter system. To construct
the reporter system, first the _sigX_ promoter region is amplified and cloned
upstream of the luc gene. Both genes are consequently expressed at the same
rate when activated by the mature transcription factor (Fig. 5).

Complying with the steps in the detection chain of _S. mutans_ , we formulate
(29)-(34) as a computational model for the _S. mutans_ -XIP bacterial sensor.
Eqs. (29) and (30) model the time evolution of the intracellular XIP amount
and extracellular XIP amount, respectively. The uptake mechanism is modeled
using an active uptake model. Eq. (31) captures the kinetics of the
intracellular $\mathsf{ComR_{f}}$ regulator which is transcribed at a basal
rate $\kappa_{\mathrm{B}}$. $\mathsf{ComR_{f}}$’s rate of production is
further up-regulated by the master transcription factor $\mathsf{SigX}$ which
is reflected by the Michaelis-Menten term in (31) [48]. As stated above, the
immature $\mathsf{ComR_{f}}$ is activated by intracellular
$\mathsf{XIP_{int}}$ with which it forms a dimer which is the mature form of
$\mathsf{ComR_{f}}$ ($\mathsf{ComR_{m}}$) that acts as TF for _sigX._ The
study on the ComRS pathway in _Streptococcus termophilus_[32] suggests that
the stoichiometry of the dimerization is such that _two ComR_ and _two XIP_
molecules are needed to produce _one_ mature $\mathsf{ComR_{m}}$ molecule.
This is reflected by the exponent $2$222It must be noted that, in general,
this exponent can also assume values different to the moieties of the
reactants.  in (29), (31) and (32), and the factor $2$ in (29) and (31). Eqs.
(33) and (34) model the gene transcription and subsequent translation
(reflected by the translation efficient parameter $\alpha_{\mathsf{sigX}}$) of
mRNA into the respective proteins, luciferase and $\mathsf{SigX}$. The
identical terms in both equations reflect that luciferase and $\mathsf{SigX}$
are produced at _identical rate._ The gene expression of both genes _sigX_ and
_lucFF_ is activated by the transcription factor $\mathsf{ComR_{m}}$ and the
corresponding transcription rate is governed by Michaelis-Menten kinetics.

As stated, the total bioluminescence response $L_{\mathrm{B}}$ of the sensor
population is directly proportional to the amount of luciferase in the system
and is thus commonly modeled as

$L_{\mathrm{B}}=\xi_{\mathsf{lum}}\mathsf{Luc}$ (28)

where $\xi_{\mathsf{lum}}$ is a proportionality constant which collects the
effects of, for example, oxygen availability, substrate concentration, etc.
The model is highly simplifying and a more precise model (e.g., [49]) would
additionally consider residual substrates which participate in the
bioluminescence reaction.

$\displaystyle\frac{\mathrm{d}\mathsf{XIP_{int}}}{\mathrm{d}t}$
$\displaystyle=\nu(t)\kappa_{\mathrm{up}}\left(\frac{\mathsf{XIP_{ext}}}{K_{\mathrm{up}}+\mathsf{XIP_{ext}}}\right)-\delta_{\mathsf{X_{int}}}\mathsf{XIP_{int}}$
$\displaystyle-2k_{\mathsf{TF_{m}}}\left(\frac{1}{\nu(t)V_{\mathrm{i}}}\mathsf{XIP_{int}}\mathsf{ComR_{f}}\right)^{2}+2k_{\mathsf{TF_{f}}}\mathsf{ComR_{m}}$
(29) $\displaystyle\frac{\mathrm{\mathrm{d}}\mathsf{XIP_{ext}}}{\mathrm{d}t}$
$\displaystyle=-\nu(t)\kappa_{\mathrm{up}}\left(\frac{\mathsf{XIP_{ext}}}{K_{\mathrm{up}}+\mathsf{XIP_{ext}}}\right)-\delta_{\mathsf{X_{ext}}}\mathsf{XIP_{ext}}$
(30) $\displaystyle\frac{\mathrm{d}\mathsf{ComR_{f}}}{\mathrm{d}t}$
$\displaystyle=\nu(t)\alpha_{\mathsf{ComR}}\kappa_{\mathrm{B}}-\delta_{\mathsf{TF}_{\mathrm{f}}}\mathsf{ComR_{f}}-2k_{\mathsf{TF}_{\mathrm{m}}}\left(\frac{1}{\nu(t)V_{\mathrm{i}}}\mathsf{XIP_{int}}\mathsf{ComR_{f}}\right)^{2}$
$\displaystyle+2k_{\mathsf{TF}_{\mathrm{f}}}\mathsf{ComR_{m}}+\nu(t)\alpha_{\mathsf{comR}}\kappa_{\mathsf{x}}\left(\frac{\left(\frac{\mathsf{SigX}}{\nu(t)V_{\mathrm{i}}}\right)^{m}}{K_{\mathsf{Sig_{X}}}^{m}+\left(\frac{\mathsf{SigX}}{\nu(t)V_{\mathrm{i}}}\right)^{m}}\right)$
(31) $\displaystyle\frac{\mathrm{d}\mathsf{ComR_{m}}}{\mathrm{d}t}$
$\displaystyle=k_{\mathsf{TF}_{\mathrm{m}}}\left(\frac{1}{\nu(t)V_{\mathrm{i}}}\mathsf{XIP_{int}}\mathsf{ComR_{f}}\right)^{2}$
$\displaystyle-
k_{\mathsf{TF}_{\mathrm{f}}}\mathsf{ComR_{m}}-\delta_{\mathsf{TF}_{\mathrm{m}}}\mathsf{ComR_{m}}$
(32) $\displaystyle\frac{\mathrm{d}\mathsf{Luc}}{\mathrm{d}t}$
$\displaystyle=\nu(t)\alpha_{\mathsf{sigX}}\kappa_{\mathsf{comR}}\left(\frac{\left(\frac{\mathsf{ComR_{m}}}{\nu(t)V_{\mathrm{i}}}\right)^{n}}{K_{\mathsf{comR}}^{n}+\left(\frac{\mathsf{ComR_{m}}}{\nu(t)V_{\mathrm{i}}}\right)^{n}}\right)-\delta_{\mathrm{L}}\mathsf{Luc}$
(33) $\displaystyle\frac{\mathrm{d}\mathsf{SigX}}{\mathrm{d}t}$
$\displaystyle=\nu(t)\alpha_{\mathsf{sigX}}\kappa_{\mathsf{comR}}\left(\frac{\left(\frac{\mathsf{ComR_{m}}}{\nu(t)V_{\mathrm{i}}}\right)^{n}}{K_{\mathsf{comR}}^{n}+\left(\frac{\mathsf{ComR_{m}}}{\nu(t)V_{\mathrm{i}}}\right)^{n}}\right)-\delta_{\mathrm{x}}\mathsf{SigX}$
(34)

####  V-C1 Parameter Estimation

Figure 6: Experimental data vs. simulation of bacterial sensor luminescence
response with fitted parameters. Experimental data is RLU data normalized with
experimental OD600 bacterial growth curves.

We proceed to fit the computational _S. mutans_ model to experimentally
obtained data by adjusting the equation system’s parameters. We consider the
_average response_ of a bacterial cell which is given by (29)-34 _discounting_
the scaling effect of the bacterial population factor $\nu(t)$. We use our
wet-lab data of the form $\text{RLU}/\text{OD600}$, i.e., luminescence data in
Relative-Light-Units (RLU) normalized with measured OD600 population density
curves (see Fig. 4). The resulting data can be taken as measure of the average
response of a viable bacterial cell in the population.

As we do not know $\xi_{\mathsf{lum}}$, estimating the parameters of the model
from the experimental data is greatly complicated. We note that
$\xi_{\mathsf{lum}}$ could experimentally be obtained by calibrating the
luminescence reader through parallel measurements of the luciferase protein
(e.g., western blotting) and comparison with the obtained RLU curves. Here, we
proceed by assuming a value $\xi_{\mathsf{lum}}=10^{4}$ which is sensible in
the context of reasonable estimation ranges for the model parameters. We
further possess only limited observations of the system (in form of the
measured luminescence as proxy for the luciferase concentration), lacking data
for the intermediary species and processes. This, too, complicates estimating
the full system parameters as they cannot be verified against data observed
for the reactions they are _directly_ linked to.

For the estimation, we restricted the optimization domain of parameters to
biologically sensible ranges. For several parameters we relied on references
in the literature as a guide for defining the estimation ranges or to fix the
parameter values in the estimation process [32].

The results of the parameter adjustment is presented in Fig. 6 and Tab. I. The
bacterial sensor system has been implemented for simulation and parameter
estimation in _COPASI_[50]. The parameter estimation was conducted using a
_Simulated Annealing_ algorithm with a start temperature of $T=1$ and a
default cooling factor of $0.85$. For our simulation and estimation we assumed
a bacterial volume of $V_{\mathrm{i}}=$1\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{3}$$ and an _effective_ extracellular volume of
$V_{\mathrm{e}}=$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{3}$$ per cell.

Table I: _S. mutans_ -XIP system parameters with adjusted values.  Parameter |  Unit | Value |  Description |  Parameter |  Unit |  Value |  Description
---|---|---|---|---|---|---|---
$\mathsf{ComR_{f}}(0)$ |  - | $17.25$ |  Initial Response Regulator Amount |  $\kappa_{\mathrm{B}}$ |  ${\mathrm{min}}^{-1}$ |  $0.492$ |  Basal comR Transcription Rate
$\kappa_{\mathrm{up}}$ |  ${\mathrm{min}}^{-1}$ | $18.35$ |  Maximal Uptake Rate |  $\kappa_{\mathsf{ComR}}$ |  ${\mathrm{min}}^{-1}$ |  $6.63$ |  Maximum ComR-induced sigX Transcription Rate
$K_{\mathrm{up}}$ |  ${\mathrm{\SIUnitSymbolMicro m}}^{-3}$ | $333.6$ |  Michaelis-Menten Constant for Active Uptake |  $K_{\mathsf{ComR}}$ |  ${\mathrm{\SIUnitSymbolMicro m}}^{-3}$ |  $14.02$ |  comR Michaelis-Menten Constant
$\delta_{\mathsf{XIP}}$ |  ${\mathrm{min}}^{-1}$ | $0.023$ |  Degradation Constant of XIP |  $\alpha_{\mathsf{SigX}}$ |  - |  $2$ |  sigX Translation Efficiency
$k_{\mathsf{TF}_{\mathrm{m}}}$ |  ${\mathrm{\SIUnitSymbolMicro m}}^{9}\text{\,}{\mathrm{min}}^{-1}$ | $0.02$ |  Transcription Factor Maturation Reaction Rate Constant |  $\kappa_{\mathrm{X}}$ |  ${\mathrm{min}}^{-1}$ |  $0.216$ |  Maximum SigX induced comR Transcription Rate
$\delta_{\mathsf{TF}}$ |  ${\mathrm{min}}^{-1}$ | $0.01$ |  Degradation Constant of ComR (fixed)[32] |  $K_{\mathsf{SigX}}$ |  ${\mathrm{\SIUnitSymbolMicro m}}^{-3}$ |  $617413$ |  sigX Michaelis-Menten Constant
$k_{\mathsf{TF}_{\mathrm{f}}}$ |  ${\mathrm{min}}^{-1}$ | $0.00001$ |  Transcription Factor De-Maturation Reaction Rate Constant |  $\delta_{\mathrm{x}}$ |  ${\mathrm{min}}^{-1}$ |  $0.59$ |  SigX Degradation Constant
$\delta_{\mathsf{TF}_{\mathrm{m}}}$ |  ${\mathrm{min}}^{-1}$ | $0.0027$ |  Degradation Constant of ComR-XIP Complex |  $\delta_{\mathrm{L}}$ |  ${\mathrm{min}}^{-1}$ |  $0.0158$ |  Luciferase Degradation Constant (fixed)[32]
$\alpha_{\mathsf{ComR}}$ |  - | $0.166$ |  comR Translation Efficiency |  $n$ |  - |  $2.81$ |  Hill Coefficient
$m$ |  - | $3.96$ |  Hill Coefficient |  |  |  |

####  V-C2 Experimental procedure

The experimental wet-lab data was obtained as follows:

##### Bacterial strains and medium

Cultures of _S. mutans_ (strain SM091 [51]) were grown in chemically defined
medium (CDM) at $5\text{\,}\mathrm{\char 37\relax}$ $\ce{CO2}$,
$37\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and stored at
$-80\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in the same medium supplemented
with $15\text{\,}\mathrm{\char 37\relax}$ glycerol for cryopreservation. CDM
is prepared as previously described [52].

##### Synthetic peptide

The _sigX_ -inducing peptide (XIP) ($\ce{NH2-GLDWWSL-COOH}$;
$98\text{\,}\mathrm{\char 37\relax}$ purity; GenScript, Piscataway, NJ, USA)
was reconstituted with $20\text{\,}\mathrm{\SIUnitSymbolMicro L}$ dimethyl
sulfoxide (DMSO) (Sigma-Aldrich, St. Louis, MI, USA), to which
$1\text{\,}\mathrm{mL}$ sterile distilled water was added to give a final
concentration of $10\text{\,}\mathrm{m\textsc{m}}$ and stored at
$-20\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Working stocks were prepared and
aliquoted at $100\text{\,}\mathrm{\SIUnitSymbolMicro\textsc{m}}$ in sterile
distilled water and stored at $-20\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

##### Luciferase reporter assays

Culture stock of _S. mutans_ reporter strain at OD600 of $0.5$ was thawed and
diluted $5$-fold in CDM. Serial dilutions of XIP were made
($2000\text{\,}\mathrm{n\textsc{m}}$, $1000\text{\,}\mathrm{n\textsc{m}}$,
$500\text{\,}\mathrm{n\textsc{m}}$, $250\text{\,}\mathrm{n\textsc{m}}$,
$125\text{\,}\mathrm{n\textsc{m}}$, $62.5\text{\,}\mathrm{n\textsc{m}}$,
$31.25\text{\,}\mathrm{n\textsc{m}}$, $15.625\text{\,}\mathrm{n\textsc{m}}$,
$7.8\text{\,}\mathrm{n\textsc{m}}$, $3.9\text{\,}\mathrm{n\textsc{m}}$,
$1.9\text{\,}\mathrm{n\textsc{m}}$ and control (No XIP added) in
$200\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of CDM aliquoted in wells of a
$96$-well microtiter plate (Thermo Fisher Scientific). Then,
$100\text{\,}\mathrm{\SIUnitSymbolMicro l}$ of previously diluted _S. mutans_
culture was added to the wells, in triplicates for each XIP concentration. A
volume of $10\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of a
$1\text{\,}\mathrm{m\textsc{m}}$ aqueous D-luciferin solution (Synchem,
Felsberg-Altenberg, Germany) was added to each well, and three wells
containing CDM and luciferin without the inoculum were used as blanks. The
plate was sealed with a Top Seal (PerkinElmer), and incubated at
$37\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $15$ hours. Relative
luminescence units (RLU) and OD600 were measured every $15$ minutes in a
multi-detection micro plate reader (Cytation 3; BioTek).

##  VI Engineering and Optimization of Bacterial Sensors

While the focus so far was on the _description_ of a generic bacterial sensor,
we now proceed to investigate strategies for altering and engineering a
bacterial sensor with specific properties. It is consequently useful to
consider general aspects of sensor analysis and engineering. Important sensor
properties are

  * •

Sensor sensitivity: Differential response of the sensor to different measurand
quantities, i.e., the difference between sensor outputs in response to
different analyte concentrations.

  * •

Response intensity: Generally, a stronger response of the bacterial sensor is
clearly desirable for the maximization of the Signal-to-Noise-Ratio (SNR).

  * •

Responsiveness: Time between first contact with analyte and _final_ sensor
response, i.e., discounting contingent overshoots during the transient
response.333This is listed here for the sake of completeness but a formal
treatment of this sensor property is not within the scope of this work.

In the following, we discuss how these general sensor properties (sensor
sensitivity and response intensity) can be evaluated in the context of the
investigated bacterial sensor models. As we aim at exploring how bacterial
sensors can be _practically_ altered and engineered for specific purposes, it
is necessary to establish a link between _parameters_ in our computational
models and _biological properties_ that are accessible to manipulation in the
wet-lab.

###  VI-A Engineering of Bacteria - Linking Model Parameters to Biological
Properties

Synthetic biology is devising ever more techniques for altering bacterial
dynamics for specific purposes, e.g., genetic or metabolic engineering. For
this paper, we are most interested in strategies that can directly change a
bacterial cell’s properties that relate to the detection chain discussed
above. We therefore briefly review a number of techniques by which bacteria
can be manipulated to evoke desired sensor dynamics. Here, we merely give a
qualitative overview. In each specific case, a quantitative relationship
between a concrete manipulation measure and the exact change in model
parameter must be established which is not within the scope of this work.

##### Regulation of Uptake

Active uptake mechanisms or phosphorylation-based internalization are governed
by cell membrane-bound ligands or channel molecules. By regulating the
expression of the genes coding for the specific permease protein one has
control of the uptake dynamics of a sensor. Considering the _S. mutans_ -XIP
__ sensor, by _up-regulating_ the expression of the _opp_ gene we could
_increase_ the uptake rate constant $\kappa_{\mathrm{up}}$ due to its
parametrization by $\mathsf{M}_{0}$ which we introduced as the number of
surface permeases per cell above.

##### Regulation of Protein Stability

The half-life of proteins involved in a biochemical system has a strong impact
on its overall dynamics. The degradation rates of proteins in bacterial cell
depend on a variety of factors: environmental conditions (e.g., pH,
temperature), the proteins structure and concentration, or post-
transcriptional modification (e.g., phosphorylation, bacterial
ubiquitination). Techniques based on __ attaching _degradation tags_ to
proteins that mark them for degradation by the cellular protein regulation
mechanisms [53] enhance the possibilities for engineering protein
concentration level in a targeted manner. Degradation tags can thus be used to
directly alter the protein degradation rate constants (see e.g.,
$\delta_{\mathsf{L}}$ in (33)).

##### Regulation of Gene Expression

Engineering of gene expression is the single most important tool in synthetic
biology. This comprises promoter engineering, plasmids which can be engineered
for specific purposes and inserted into bacteria, and the targeted knocking
out (see _comS_ in _S. mutans_ -XIP __ model above) or insertion of genes
(e.g., promoter genes) directly into the bacterial genome. The number of
plasmids inserted into bacterial cells, for example, directly modulates the
expression levels of the genes on the plasmid as more promoter sites are
available for binding transcription factors. According to Appendix Sec. A, the
number of promoter sites thus directly modulates the transcription rate (e.g.,
$\kappa_{\mathsf{comR}}$ in (33)). Considering the _S. mutans-XIP_ sensor
example discussed above,__ this can be used to regulate transcription of the
_sigX-lucFF_ operon, or to control $\kappa_{\mathrm{B}}$, i.e., the basal
transcription rate of the response regulator. This facilitates a finer
regulation of the modulation effect of the response regulator.

###  VI-B Steady-State Optimization

Figure 7: RRE- & S-system model numerical solution comparison.

Using the introduced S-system formalism, we construct the S-system
representation based on a RRE model of a generic bacterial sensor following
the same structure as the discussed _S. mutans_ -XIP model and investigate it
with respect to the sensor properties listed above. This section treats the
abundant-analyte case (corresponding to _Scenario 1_ introduced in Sec. II).
The parameter values of the RRE model used in this section are chosen somewhat
arbitrary without reference to an actual bacterial sensor but we assumed
realistic values for each parameter. For the transcription factor maturation
reaction we assumed moieties of $1$ for both reactants but allowed for non-
unity reaction orders $N$ and $M$.

#### RRE Model

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{S_{in}}$
$\displaystyle=\kappa_{\mathrm{up}}\,\left(\frac{\mathsf{S_{ext}}}{K_{\mathrm{up}}+\mathsf{S_{ext}}}\right)+k_{\mathsf{TF}_{\mathrm{f}}}\,\mathsf{TF}$
$\displaystyle-
k_{\mathsf{TF}_{\mathrm{m}}}\,\mathsf{\mathrm{RR}}^{M}\,\mathsf{S_{\mathrm{in}}^{N}}-\delta_{\mathsf{S}}\,\mathsf{S_{\mathrm{in}}}$
(35) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{RR}$
$\displaystyle=\underset{\text{Basal
Transcription}}{\alpha_{\mathsf{RR}}\underbrace{\kappa_{\mathsf{RR}}}}+k_{\mathsf{TF}_{\mathrm{f}}}\,\mathsf{TF}$
$\displaystyle-
k_{\mathsf{TF}_{\mathrm{m}}}\,\mathsf{RR}^{M}\,\mathsf{S}_{\mathsf{in}}^{N}-\delta_{\mathsf{RR}}\,\mathsf{RR}$
(36) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{TF}$
$\displaystyle=k_{\mathsf{TF}_{\mathrm{m}}}\,\mathsf{RR}^{M}\,\mathsf{S}_{\mathsf{in}}^{N}$
$\displaystyle-
k_{\mathsf{TF}_{\mathrm{f}}}\,\mathsf{TF}-\delta_{\mathsf{TF}}\,\mathsf{TF}$
(37) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{P}$
$\displaystyle=\frac{\mathrm{\alpha}\,\kappa_{\mathrm{max}}\,\mathsf{TF}^{n_{\mathrm{Hill}}}}{K^{n_{\mathrm{Hill}}}+\mathsf{TF}^{n_{\mathrm{Hill}}}}-\delta_{\mathsf{P}}\,\mathsf{P}$
(38)

#### S-System Model

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{S}_{\mathsf{in},S}$
$\displaystyle=\alpha_{\mathsf{S}}\,\mathsf{TF}_{\mathsf{S}}^{g_{\mathsf{S,\mathsf{TF}}}}\,\mathsf{S}_{\mathsf{ext},S}^{g_{\mathsf{S_{ext,}S}}}$
$\displaystyle-\beta_{\mathsf{S}}\,\mathsf{RR}_{S}^{h_{\mathsf{S,\mathsf{RR}}}}\,\mathsf{S}_{\mathsf{in},S}^{h_{\mathsf{S,S}}}$
(39) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{RR}_{S}$
$\displaystyle=\alpha_{\mathsf{RR}}\,\mathsf{TF}_{S}^{g_{\mathsf{RR},\mathsf{TF}}}$
$\displaystyle-\beta_{\mathsf{RR}}\,\mathsf{RR}_{S}^{h_{\mathsf{RR,RR}}}\,\mathsf{S}_{\mathsf{in},S}^{h_{\mathsf{RR,S}}}$
(40) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{TF}_{S}$
$\displaystyle=\alpha_{\mathsf{TF}}\,\mathsf{RR}_{S}^{g_{\mathsf{TF,RR}}}\,\mathsf{S}_{\mathsf{in},S}^{g_{\mathsf{TF,S}}}$
$\displaystyle-\beta_{\mathsf{\mathsf{TF}}}\,\mathsf{TF}_{S}^{h_{\mathsf{TF,TF}}}$
(41) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{P}_{S}$
$\displaystyle=\alpha_{\mathsf{P}}\,\mathsf{RR}_{S}^{g_{\mathsf{P,RR}}}\,\mathsf{TF}_{S}^{g_{\mathsf{P,\mathsf{TF}}}}$
$\displaystyle-\beta_{\mathsf{P}}\,\mathsf{P}_{S}^{h_{\mathsf{P,P}}}$ (42)

(a)

(b)

Figure 8: S-system steady-state sensitivity analysis. (a) Effect of parameter
alterations on S-system steady-state solutions. The center graph group
represents the base system with the initial parameter set. (b) Sensitivities
with respect to model parameters.

Eqs. (35)-(38) are the RRE model of the bacterial sensor and (39)-(42)
represent the corresponding S-system derived according to (4) and (5). The
operating points of the S-system representation
$\mathbf{X}_{0}=\left[\mathsf{S}_{\mathsf{in},0},\mathsf{RR}_{0},\mathsf{TF}_{0},\mathsf{P}_{0}\right]^{T}$
are chosen so that the steady-state solutions of both systems are near
identical. The natural choice of $\mathbf{X}_{0}$ are the _steady-state
solutions_ of the RRE system. Solving the equation systems numerically yields
solution trajectories as plotted in Fig. 7. Note that the transient responses
of the S- and RRE-system representations can diverge significantly while the
steady-state solutions are in general identical.

The S-system representation can then be analytically solved for steady-state
as in (6) and the log-gain matrix $\mathbf{L}$ obtained explicitly. Solving
repeatedly for the steady-state bacterial sensor output $\mathsf{P}_{SS}$ with
different values of the external/independent variable $\mathsf{S_{ext}}$ and
plotting in the log-log space yields a line whose slope reflects the log gain
of the sensor (Fig. 8a). The S-system used to generate the data is constructed
around the numerically obtained RRE-system steady state solution for the mid-
value of the tested external variable $\mathsf{S_{ext}}$ range. This reflects
that a single S-system representation can accurately capture the system
dynamics responding to a wider range of input values.

Next, the log-gain sensitivities $S_{L_{\mathsf{P}},\theta_{k}}$ and
sensitivities $S_{m_{\mathsf{P}},\theta_{k}}$ are obtained using (7) and (8)
to measure the relative change in the log-gain and the constant component of
the steady-state sensor response in reaction to a relative change in a RRE
model parameter $\theta_{k}$. Importantly, and considering the above mentioned
sensor properties, $S_{L_{j},\theta_{k}}$ and $S_{m_{j},\theta_{k}}$
constitute measures for _sensor sensitivity_ and _response_ _intensity_ ,__
respectively.__ The results of this sensitivity analysis are shown in Fig. 8b.

Noticeably, we find that $S_{L_{\mathsf{P}},\theta_{k}}$ and
$S_{m_{\mathsf{P}},\theta_{k}}$ have opposite signs for a number of evaluated
RRE parameters. This suggests that changes in several RRE system parameters
have _diametric_ effects on _sensor sensitivity_ and _response_ _intensity._
In other words, according to this analysis _increasing_ the steady-state
response strength of a bacterial sensor by manipulating a manipulable subset
of these parameters (via techniques alluded to above) _diminishe_ s its
sensitivity and vice versa.

###  VI-C Dynamic Sensitivity Analysis

While the preceding section investigates the _steady-state_ response of a
bacterial sensor, we now examine the case when the signal analyte is scarce
(i.e., non constant; see _Scenario 2_ described in Sec. II) and either is
depleted naturally or enters the bacterial sensor system dynamics as a state
variable without a compensating influx. We are interested in how the findings
from the steady-state analysis map onto the time-course analysis of the sensor
response.

For the dynamic sensitivity analysis, $\mathsf{S_{ext}}$ cannot be treated as
external variable and enters the equation system (35)-38 as additional
equation as (see also (30))

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathsf{S_{ext}}=$
$\displaystyle-\kappa_{\mathrm{up}}\,\left(\frac{\mathsf{S_{ext}}}{K_{\mathrm{up}}+\mathsf{S_{ext}}}\right)-\delta_{\mathsf{S}}\,\mathsf{S_{\mathrm{ext}}}.$
(43)

Here we work with the RRE system directly as our focus is on the transient
response. In particular, we isolate the _maximum response_ as the point of
interest as is common in the related literature. As above, we investigate the
influence of model parameters on both sensor sensitivity and sensor signal
intensity.

Response intensity is investigated by appending one set of differential
equations (9) per investigated parameter to the ODE system (35)-38 and solving
the resulting system numerically with an appropriate solver (we use the ode15s
solver in Matlab). Thereby, trajectories as in Fig. 9a are obtained which are
then sampled corresponding to the point of maximum transient response of the
sensor output $\mathsf{P}$, i.e., in the time-course solution of the sensor
response, the time of maximum response $\tau_{\max\mathsf{P}}$ is extracted
and $S_{\mathsf{P},\theta_{k}}(\tau_{\max\mathsf{P}})$ evaluated.

The entire dynamic sensitivity trajectory of the variable of interest with
respect to a specific parameter predicts how a change in the parameter affects
the time-course of the variable. For instance, the dynamic sensitivity for the
parameter $n_{\mathrm{Hill}}$ predicts well how changes in it affect
$\mathsf{P}$ (Fig. 9b).

As described in Sec. III-D, we can compute the dynamic sensitivity with
respect to the amount of a species (state-variable) at $t=0$ (i.e., the
initial conditions of the corresponding ODE system). Doing so for $\mathsf{P}$
with respect to $\mathsf{S_{ext}}(0)$, i.e., calculating
$S_{\mathsf{P},\mathsf{S_{ext}(0)}}=\frac{\partial\mathsf{P}}{\partial\mathsf{S_{ext}}(0)}$,
yields the results shown in Fig. 10 where the sampled value, i.e.,
$S_{\mathsf{P},\mathsf{S_{ext}(0)}}(\tau_{\max\mathsf{P}})$, corresponds well
to the slope of the line in the inset plot. The latter shows the maximum
response of $\mathsf{P}$ evaluated for several values of $\mathsf{S_{ext}}(0)$
in log-log space.

Equivalent to the steady-state log-gain sensitivities, we then proceed to
investigate the impact of model parameters on
$S_{\mathsf{P},\mathsf{S_{ext}(0)}}$ by solving (11) for all
$\theta_{k}\in\mathbf{\Theta}_{R}$. The corresponding results are presented in
Fig. 11.

Comparing Fig. 8b and Fig. 11b reveals that the sensitivities for steady-state
and transient maximum exhibit notable differences. While, for instance,
evaluating $S_{L_{\mathsf{P}},\kappa_{\mathsf{RR}}}$, $\frac{\partial
S_{\mathsf{P},\mathsf{S_{ext}(0)}}}{\partial\kappa_{\mathsf{RR}}}$ and
$S_{\mathsf{P},\mathsf{\kappa_{\mathsf{RR}}}}$ implies that a positive change
in $\kappa_{\mathsf{RR}}$ has only a minor effect on the steady-state log-gain
(namely _decreasing_ it), the maximum transient response experiences both a
positive shift in magnitude _and_ a minor increase in sensitivity (as is also
visualized in Fig. 11a). Other trends observed in the stead-state analysis
are, in contrast, reproduced by the dynamic sensitivity analysis; alterations
in $n_{\mathrm{Hill}}$, for example, elicit strong diametric effects on signal
intensity and sensitivity for both cases.

The sensitivity analysis results for both steady-state and transient response
presented in this section have validity in this form only for the arbitrary
parameter set alluded to above. However, the general findings of the
investigation could be confirmed for various tested parameter sets.

(a)

(b)

Figure 9: Dynamic Sensitivity Analysis. (a) Relative Dynamic Time-Course
Sensitivities. Red circles mark evaluation at maximum response of
$\mathsf{P}$. (b) Sensor response over range of values of $n_{\mathrm{Hill}}$.
Figure 10: Dynamic Sensitivity with respect to $\mathsf{S_{ext}}(0)$.

(a)

(b)

Figure 11: Dynamic Sensitivity Analysis Results. (a) Effect of parameter
alterations. The blue line represents the base system with unaltered
parameters. (b) Sensitivities for model parameters.

##  VII Conclusion

In this work we systematically investigated the quantitative properties of the
biochemical mechanisms governing bacterial sensor systems. We dissected the
bacterial sensor detection chain and provided an overview of models for
describing them. Based on this, we presented a _S. mutans_ -XIP __ sensor as a
case-example for which we fitted the derived model to experimental wet-lab
data. We obtained a good match between our computational model and our
empirical data yielding biologically meaningful parameter values which are in
the expected ranges. While no general validity of the fitted parameter values
can be guaranteed, the results demonstrate the applicability of the
methodological approach. Our mathematical modeling approach is limited to
deterministic models, which is adequate for the scope of this paper, but may
be insufficient to study the inherent stochasticity of molecular systems in
greater detail. Further, we applied rather restrictive assumptions (aqueous
assays, homogeneous analyte distribution, homogeneous expression levels of
bacterial sensor cells) to the analysis. These shortcomings may be addressed
in future works.

The main contribution of this paper is the analytic and numerical derivation
of common sensor-engineering measures, e.g., the _sensor sensitivity,_ for
bacterial sensors systems. This was achieved by rigorous application of BST
methods which yielded analytic results for the steady-state treatment of
bacterial sensors. We compared these findings with results obtained from a
dynamic time-course analysis where we focused on the maximum response of a
bacterial sensor. Specifically, we mathematically investigated how various
parameters from the computational models influence log-gain (sensor
sensitivity) and response strength levels for both steady-state and maximum
sensor response. We found that, in many cases, with notable exceptions,
parameter changes have opposite effects on bacterial sensor sensitivity and
response intensity levels for both steady-state and maximum transient
response. This implies a trade-off which must be taken into account when
designing bacterial sensor systems. We mentioned the _responsiveness_ of a
sensor as crucial property but did not further investigate it in this work;
this might be addressed in a future study.

We further identified and discussed a selection of parameters that are most
accessible for manipulation from a bio-engineering perspective. A systematic
investigation that links concrete manipulations in the wet-lab to numerical
changes in model parameters may be considered for future research.

By combining biological expertise with multifaceted computational techniques,
we aimed for complementing research in MC with a focus on the quantitative
aspects of engineering and optimizing bacterial sensing systems. We expect
that this work will contribute to guiding cross-disciplinary researchers who
want to computationally investigate bacterial sensors.

##  Appendix A Modeling Gene Transcription

For the sake of completeness, the dynamics of gene expression in a bacterial
sensor are summarized and a well-established mathematical representation is
derived (see e.g., [12, 28]):

  * •

Transcription factor $\mathsf{TF}$__ binding to promoter $\mathsf{Pr}$ of a
gene __ is reflected by the reaction equation

$\ce{n\mathsf{TF}+\mathsf{Pr}<=>[k_{\mathrm{a}}][k_{\mathrm{d}}]\mathsf{TF}_{n}\ce{\mathsf{Pr}}}$
(44)

where $k_{\mathrm{a}}$ and $k_{\mathrm{d}}$ denote the forward and backward
reaction rate constants. Here, $n$ defines the affinity of the binding process
and denotes the number of transcription factor molecules binding to a promoter
site, thereby activating it and increasing its affinity for binding polymerase
and thus initiating transcription.

  * •

Transcription of a gene into mRNA $\mathsf{p}_{\mathrm{mRNA}}$ follows the
reaction equation

$\ce{\mathsf{TF}_{n}\mathsf{P}->[k_{\mathrm{r}}]\mathsf{p}_{\mathrm{mRNA}}->[\delta_{\mathrm{r}}]\emptyset}$
(45)

with transcription reaction rate constant $k_{\mathrm{r}}$ and decay rate
constant $\delta_{\mathrm{r}}$.

  * •

Translation of mRNA into protein P with rate constant $k_{\mathrm{T}}$ is
reflected by

$\ce{\mathsf{p}_{\mathrm{mRNA}}->[k_{\mathrm{T}}]\mathsf{P}}$ (46)

It is practical to exploit the different time-scales of promoter-TF binding
and transcription, and the constant number of promoter sites to derive a
compact expression for the process. Consequently, the transcription dynamics
are conveniently described by Michaelis-Menten or Hill kinetics where the
former is a special case of the latter and assumes a Hill coefficient of
$n_{\mathrm{Hill}}=1$. $n_{\mathrm{Hill}}$ conceptually corresponds to the
parameter in (44) where $n_{\mathrm{Hill}}$ can assume non-integer values.
Consequently, transcription can be modeled as,

$\frac{\mathrm{d}\mathsf{p}_{\mathrm{mRNA}}}{\mathrm{d}t}=\kappa_{\mathsf{T}}\left(\frac{\mathsf{TF}^{n_{\mathrm{Hill}}}}{K^{n_{\mathrm{Hill}}}+\mathsf{TF}^{n_{\mathrm{Hill}}}}\right)-\delta_{\mathrm{r}}\mathsf{p}_{\mathrm{mRNA}}$
(47)

with $K=\frac{k_{\mathrm{d}}+k_{\mathrm{r}}}{k_{\mathrm{a}}}$and
$\kappa_{\mathsf{T}}=\mathsf{Pr}_{0}k_{\mathrm{r}}$.

The mRNA is then translated into protein with translation rate constant
$k_{\mathrm{l}}$ following

$\frac{\mathrm{d}\textsf{P}}{\mathrm{d}t}=k_{\mathrm{l}}\mathsf{p}_{\mathrm{mRNA}}-\delta_{P}\mathsf{P}.$
(48)

With the quasi-steady-state assumption for $\mathsf{p}_{\mathrm{mRNA}}$ (based
on mRNA being in general much shorter lived relative to the corresponding
protein which allows for the quasi-steady-state assumption for
$\mathsf{p}_{\mathrm{mRNA}}$), solving (47) for $\mathsf{p}_{\mathrm{mRNA}}$
in steady state and inserting into (48) gives

$\frac{\mathrm{d}\mathsf{P}}{\mathrm{d}t}=\alpha_{\mathrm{T}}\kappa_{\mathsf{T}}\left(\frac{\mathsf{TF}^{n}}{K^{n}+\mathsf{TF}^{n}}\right)-\delta_{\mathsf{P}}\mathsf{P}$
(49)

with $\alpha_{\mathrm{T}}=\frac{k_{l}}{\delta_{b}}$ as the _translation
efficiency._ Eq. (49) can further include a constant term
$\kappa_{\mathrm{B}}$ which denotes the _basal transcription rate_ at which
DNA is transcribed in the absence of up-regulation.

##  Appendix B Derivation of Dynamic Log-Gain Sensitivity Equation

For the derivation of (11) we extend the argumentation resulting in (9) [38].
Eqs. (50a) \- (50f) outline the derivation. In (50b) the identity
$\frac{\partial\dot{X}_{i}}{\partial X_{j}(0)}=0$ is attributed to $X_{j}(0)$
not occurring explicitly in $\dot{X}_{i}$.

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial
S_{i,X_{j}(0)}}{\partial\theta_{k}}\right)$
$\displaystyle=\frac{\mathrm{\partial}}{\mathrm{\partial}\theta_{k}}\left(\frac{\mathrm{d}S_{i,X_{j}(0)}}{\mathrm{d}t}\right)$
(50a)
$\displaystyle=\frac{\mathrm{\partial}}{\mathrm{\partial}\theta_{k}}\Biggl{(}\underset{=0}{\underbrace{\frac{\partial\dot{X}_{i}}{\partial
X_{j}(0)}}}\frac{X_{j}(0)}{X_{i}}+\underset{=S_{i,X_{j}(0)}\frac{X_{i}}{X_{j}(0)}}{\underbrace{\frac{\partial
X_{i}}{\partial
X_{j}(0)}}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{X_{j}(0)}{X_{i}}\right)\Biggr{)}$
(50b)
$\displaystyle=\frac{\mathrm{\partial}}{\mathrm{\partial}\theta_{k}}\left(\frac{\partial
X_{i}}{\partial
X_{j}(0)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{X_{j}(0)}{X_{i}}\right)$ (50c)
$\displaystyle=\frac{\mathrm{\partial}}{\mathrm{\partial}\theta_{k}}\left(S_{i,X_{j}(0)}\frac{X_{i}}{X_{j}(0)}\right)\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{X_{j}(0)}{X_{i}}\right)+\frac{\partial
X_{i}}{\partial
X_{j}(0)}\frac{\mathrm{\partial}}{\mathrm{\partial}\theta_{k}}\left(\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{X_{j}(0)}{X_{i}}\right)\right)$
(50d) $\displaystyle=\left(\frac{\partial
S_{i,X_{j}(0)}}{\partial\theta_{k}}X_{i}+S_{i,X_{j}(0)}\frac{\partial
X_{i}}{\partial\theta_{k}}\right)\frac{\mathrm{d}}{\mathrm{d}t}X_{i}^{-1}+X_{j}(0)\frac{\partial
X_{i}}{\partial X_{j}(0)}$ (50e) $\displaystyle=-\left(\frac{\partial
S_{i,X_{j}(0)}}{\partial\theta_{k}}X_{i}+S_{i,X_{j}(0)}\frac{\partial
X_{i}}{\partial\theta_{k}}\right)\frac{\dot{X_{i}}}{X_{i}^{2}}+X_{j}(0)\frac{\partial
X_{i}}{\partial
X_{j}(0)}\left(2X_{i}^{-3}\dot{X_{i}}-X_{i}^{-2}\frac{\partial\dot{X_{i}}}{\partial\theta_{k}}\right)$
(50f)

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