Towards Autonomous Control of Molecular
Communication in Populations of Bacteria
Chieh Lo
Guopeng Wei
Radu Marculescu
Carnegie Mellon University
Pittsburgh, PA, 15213
Carnegie Mellon University
Pittsburgh, PA, 15213
Carnegie Mellon University
Pittsburgh, PA, 15213
chiehl@andrew.cmu.edu
guopengw@ece.cmu.edu
radum@cmu.edu
ABSTRACT
Quorum sensing is a chemical communication process bacteria use to sense population density and regulate their collective behavior. By using quorum sensing inhibitors that
degrade quorum sensing molecules and inactivate their receptors, one can inhibit bacterial pathogenesis, such as the
production of virulence and biofilm development. To keep
the level of quorum sensing molecules below the activation
threshold, we propose a biological controller that can generate different concentration levels of inhibitors under different environment conditions. We provide a detailed analysis
of our proposed controller that includes system response,
stability, and sensitivity analysis. We also discuss the autonomous controller design under specified environment constraints and validate our results via simulation. This work
represents a first step towards a paradigm change in reducing bacterial pathogenesis via controlling the dynamics of
bacterial cell-to-cell communication network.
1.
INTRODUCTION
Cell-to-cell communication [1][4][16] mechanism known as
quorum sensing (QS) [3] utilizes cell density information to
monitor the environment and alter gene expressions in a
competitive environment. The QS system used by gramnegative bacteria (see Figure 1(a)) is mediated by diffusible
signaling molecules that belong to the acyl-homoserine lactone (AHL) family [3]. By sensing the concentration of specific types of the AHL molecules, bacteria can form biofilms,
express virulence, and become resistant to antibiotics after
reaching a high cell density [11].
In this paper, we aim at reducing the bacterial pathogenesis [14] by quenching the cell-to-cell bacterial communication, or more specifically, by keeping the concentration of the
signaling molecules under a strain-specific activation threshold. Due to high mutation rates, bacteria can develop resistance to antibiotics quickly [17]. QS inhibitors (QSIs) [12]
can block the virulence regulatory pathways without killing
or affecting bacteria growth [13]. However, QSI resistant
mutants which can continue to utilize QS communication
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have a better chance of survival. To prevent the selective
pressure caused by QSIs [5], multi-inhibitor schemes can be
adopted. However, synthesizing multiple inhibitors at the
same time and in large quantities can not only result in
wasting chemical substances but also produce toxic effects
to the host [10]. Therefore, in this paper, we propose to
design a biological controller which can generate the minimal amount of QSIs that can stop bacteria communication
without inducing much selective pressure.
To synthesize the QSIs and control the concentration of
the AHL molecules [9], genetic circuits can serve as the basic
control units which automatically detect and react to the
environment changes. For example, we can construct genetic
circuits by cloning the genes in the plasmid, such as the
aiiA gene which expresses the enzyme that hydrolyzes the
AHL molecules [20]. However, due to the uncertainty in the
biological parameters (for example, the values of the kinetic
constants) of the genetic parts [2], such systems may be
unable to repress the QS regulation system at all. Therefore,
stability and sensitivity analysis are needed to demonstrate
the feasibility of the proposed system.
Instead of implementing the genetic circuit directly within
live bacteria which has potential for side effects, we propose
to use artificial cells (see Figure 1(b)). For instance, we
can engineer the permeability of the cell membrane [18] via
paper-based techniques as recently proposed in [19]. Based
on this new technology, it becomes possible to construct
artificial cells by cloning the genome sequence within the
genes of the bacteria host and using liposome as the cell
membrane. By placing the artificial cells into the bacterial
environment, it becomes possible to detect the concentration
of the AHL molecules and then generate the right amount of
QSIs in real-time. Consequently, the proposed system represents a paradigm shift from manual to autonomous control
of bacteria communication network.
In the following sections, we target the opportunistic human pathogen Pseudomonas aeruginosa which has been identified as a primary target for antimicrobial research [22].
Taken together, our contribution is threefold:
• First, we develop an ordinary differential equation (ODE) based model of Pseudomonas aeruginosa QS regulation system and propose the synthetic circuitry to
be used in the artificial cell. To the best of our knowledge, this is the first design that formally considers the
autonomous control of the QS signal.
• Second, we formulate the control problem and perform
the stability and sensitivity analyses. This provides
the theoretical basis for synthesizing the controller.
• Third, we identify the feasible regions of operation
(a)
(b)
P. aeruginosa
&'($!
&'($!
(c)
Artificial Cell
LasR inhibition
Input Signal (d(t))
'"#$! '"#$!
&'(%!
Plant
Bacteria
(X(t))
#'(%!
#'($!
"%%&!
!"#$!
$"#$!
$"#$!
$"#$!
$"#$!
AI
AI2
Antagonist
AiiA
$"#%!
!"#$!
Reference Signal (r(t))
&'($!
!"#%!
AHL inhibition
"!
!!
Process Variable
(y(t))
Controller
Artificial Cell
(U(t))
Figure 1: Proposed Control System: QS Regulation System in P. aeruginosa and Proposed Controller Circuit. (a) The QS
regulation system of P. aeruginosa. (b) The artificial cell contains genetic circuitry. The aiiA genes can produce the AiiA enzyme
that hydrolyzes the AHL molecules. The black rectangle box produces the triangle proteins which function as the antagonists
of LasR. The orange circle represents AI that moves freely through the cell membrane, the green diamond represents AI2 and
the purple and black particle are AiiA and antagonist molecules, respectively. The red, orange, green and purple oval represent
LasR, LasI, RhlR and RhlI, respectively. (c) The diagram of the proposed control system.
and characterize the biological parameters for minimal expression of inhibitors. We verify our proposed
controller via numerical methods and simulation. As
such, the design procedure we provide can be used as a
general guideline towards in vitro construction of synthetic cellular controllers.
d[C1 ]
dt
d[C2 ]
dt
d[C3 ]
dt
d[R1 ]
dt
The remainder of this paper is organized as follows. Section 2 focuses on the mathematical modeling of the QS regulation system of P. aeruginosa and the QSIs. Section 3
discusses the genetic circuits needed to express QSIs and incorporates the QS regulation system to form an autonomous
control system. Section 4 analyzes the system response, stability, and sensitivity via numerical methods. Section 5 provides a design example and some general design guidelines.
Section 6 utilizes the bacterial simulator proposed in [23] to
validate the model in scenarios that mimic realistic settings.
Finally, some conclusions are drawn in section 7.
2.
2.1
MATHEMATICAL MODEL
QS Model
As shown in Figure 1(a), two interconnected circuits in P.
aeruginosa form a QS cascade where the Las system sits
at the top. LasI and RhlI synthases synthesize two different kinds of AHL molecules: 3-oxo-C12-homoserin lactone (3OC12HSL) and butanoyl homoserin lactone (C4HSL), respectively1 . The AHL molecules can passively diffuse through bacterial cell membrane and bound to the LasR
and RhlR receptors in the intracellular space. The LasRAI complex (C1 ) and RhlR-AI2 complex (C2 ) activate the
transcriptions of numerous genes including not only the virulence genes, but also the downstream lasI and rhlI genes
which form the autocatalytic loop. Therefore, each cell must
dump its production of AHL molecules in the extracellular
space; otherwise, the accumulation of the intracellular AHL
molecules will induce the QS regulation system.
As the cell density increases, the diffusion process becomes
less effective and the autocatalytic loop gets enabled; this
results in the expression of QS signals. By adopting the
model in [7] and introducing the cell density factor (ρ) [6],
the following ODEs describe the QS regulation system of P.
aeruginosa:
1
Denoted as AI and AI2 in this paper
= α1 [R1 ][A1 ] − δ1 [C1 ]
(1)
= α2 [R2 ][A2 ] − δ2 [C2 ]
(2)
= α3 [R2 ][A1 ] − δ3 [C3 ]
(3)
= −α1 [R1 ][A1 ]+δ1 [C1 ]−bR1 [R1 ]+
VR1 [C1 ]
KR1 +[C1 ]
+R10 (4)
VR2 [C1 ]
d[R2 ]
= −α2 [R2 ][A2 ]+δ2 [C2 ]−bR2 [R2 ]+
+R20 (5)
dt
KR2 +[C1 ]
d[A1 ]
= −α1 [R1 ][A1 ] + δ1 [C1 ] − bA1 [A1 ]
dt
VA1 [C1 ]
d1
+
([A1EX ] − [A1 ]) + A10
+
KA1 + [C1 ]
ρ
d[A2 ]
= −α2 [R2 ][A2 ] + δ2 [C2 ] − bA2 [A2 ]
dt
VA2 [C2 ]
d2
+
([A2EX ] − [A2 ]) + A20
+
KA2 + [C2 ]
ρ
(6)
(7)
d[A1EX ]
d1
=−
([A1EX ] − [A1 ]) − bA1EX [A1EX ]
dt
1−ρ
(8)
d[A2EX ]
d2
=−
([A2EX ] − [A2 ]) − bA2EX [A2EX ]
dt
1−ρ
(9)
where [X] denotes the concentration of a particular molecular species X. C1 : LasR-AI, C2 : RhlR-AI, C3 : RhlR-AI2,
A1 : intracellular AI, A2 : intracellular AI2, A1EX : extracellular AI, A2EX : extracellular AI2, R1 : LasR, R2 : RhlR,
α: binding rate constant, δ: dissociation rate constant, b:
degradation rate constant, A10 , A20 , R10 , R20 : basal production rate constant, V : maximum production rate constant,
K: Michaelis-Menten constant, d: diffusion constant.
To give some intuition, Eqs.(1)-(3) describe the binding
reaction of the AHL molecules and LasR receptors. We use
quasi-steady state approximation to account for the protein
production rate of Eqs.(4)-(7) (the forth term) which is governed by the Michaelis-Menten kinetics. Since the cell membrane diffusion becomes less efficient when the cell density
increases, the effect can be modeled as the division of diffusion coefficient over cell density in Eqs.(6)-(9).
2.2
QS Inhibitors (QSIs)
The QSIs are designed for different targets. Some QSIs
target the LasR receptors through the use of the AHL analogues that act as antagonists for the native AHL molecules.
Due to the high affinity in binding with LasR, the antagonist molecules can out-compete the native AHL molecules
and inhibit the transcription of both LasR and LasI. Another kind of QSIs target the AHL molecules that diffuse in
the extracellular space. They hydrolyze the bond between
the acyl tail and lactone ring of the AHL molecules in a
nonreversible manner; this can directly decrease the concentration of the AHL molecules thus reducing virulence.
2.2.1
LasR/RhlR inhibitors
To model the effect of LasR type QSIs, we need to add additional equations to the QS regulation system of P. aeruginosa [8]. To begin with, the LasR antagonist molecules can
bind to the LasR and RhlR and form the complexes C4 and
C5 which directly affect the production rates of R1 and R2 :
d[C4 ]
= α4 [R1 ][A1A ] − δ4 [C4 ]
dt
d[C5 ]
= α5 [R2 ][A1A ] − δ5 [C5 ]
dt
d[R1 ]
= −α1 [R1 ][A1 ] + δ1 [C1 ] − bR1 [R1 ]
dt
VR1 [C1 ]
+
+ R10 − α4 [R1 ][A1A ] + δ4 [C4 ]
KR1 + [C1 ]
d[R2 ]
= −α2 [R2 ][A2 ] + δ2 [C2 ] − bR2 [R2 ]
dt
VR2 [C1 ]
+
+ R20 − α5 [R2 ][A1A ] + δ5 [C5 ]
KR2 + [C1 ]
(10)
(11)
(12)
(13)
d[A1AE ]
= −d3 ([A1AE ] − [A1A ]) − bA1AE [A1AE ]
dt
(14)
(15)
where A1AE : extracellular antagonist molecules.
2.2.2
d[E1EX ]
(18)
= −d4 ([E1EX ] − [E1 ]) − bE1EX [E1EX ]
dt
2
d[E2 ]
1
[C1 ]
= PlasRE (
+ 2
)−bE2 [E2 ]+d5 ([E2EX ]−[E2 ])
2 rE2
dt
KE2 +[C1 ]2
(19)
The AHL inhibitor hydrolyzes the extracellular AHL molecules which can be viewed as a degradation source and assumed to follow the Michaelis-Menten kinetics. Accordingly,
Eq.(8) should be modified as
(16)
where Enzyme denotes the extracellular AHL inhibitor.
3. CONTROL SYSTEM DESIGN
3.1 Genetic Circuit Design and Modeling
The desired functionalities of the artificial cells need to
provide stable control mechanisms to ensure that the concentration of the AHL molecules remains below the autocatalytic activation threshold. We now only consider AI
(3OC12HSL) as the primary control targeted molecules since
the activation of Rhl needs a high concentration of AI.
The genetic circuit needs to perform two main functions,
namely, detection and reaction. The detection of the AI
concentration can be realized with a specific type of protein
that can bind to the AI. In this paper, we use the same receptor protein, LasR, to detect the concentration of the AI
since it can truly experience the same concentration of the
(20)
where E1 : intracellular AI inhibitor, E2 : intracellular
LasR/RhlR inhibitor, E1EX : extracellular AI inhibitor, E2EX :
extracellular LasR/RhlR inhibitor.
The production rate (second term in Eqs. (17) and (19)) of
the inhibitors can be characterized by the cooperation binding of the LasR-AI complex which can be modeled by the
hill equation with hill coefficient n [2]. The product of the
promoter strength (PlasR ) and the basal production rate (r)
characterizes the minimal expression rate when there is no
LasR-AI complex present. We assume that the membrane
permeability of the enzyme produced by the genetic circuit
is the same as the AHL molecules.
3.2
AHL inhibitors
d[A1EX ]
= −d1 ([A1EX ]−[A1 ])−bA1EX [A1EX ]
dt
VE1 [Enzyme][A1EX ]
−
KA1EX +[A1EX ]
1
[C1 ]2
d[E1 ]
= PlasRE (
+ 2
)−bE1 [E1 ]+d4 ([E1EX ]−[E1 ])
1 rE1
dt
KE1 +[C1 ]2
(17)
d[E2EX ]
= −d5 ([E2EX ] − [E2 ]) − bE2EX [E2EX ]
dt
where A1A : intracellular antagonist molecules, C4 : LasRAntagonist and C5 : RhlR-Antagonist.
The last two terms in (12) and (13) describe the degradation and dissociation result from the antagonists. We assume that the antagonists follow the same rules as the AHL
molecules (Eqs.(6)-(9)) which can be formulated as follows:
d[A1A ]
= −α4 [R1 ][A1A ] + δ4 [C4 ] − α5 [R2 ][A1A ]
dt
+ δ5 [C5 ] − bA1A [A1A ] + d3 ([A1AE ] − [A1A ])
AI as in the bacteria. After detecting a specific concentration of the LasR-AI complex, the genetic circuit reacts to it
and produces the desired amount of inhibitors.
As described in Section 2, two types of inhibitors are used
to regulate the concentration of the AI. To obtain variable
combinations of the inhibitors with minimal expression levels that can repress the QS communication, we build two
circuits separately and set different expression levels given
different LasR-AI concentrations. Circuit I is built upon assembling the aiiA protein generator with the las promoter.
Similarly, Circuit II is built with genes that can express the
antagonists to bind with the LasR and RhlR protein (Figure 1(b)). The protein expression mechanism of the genetic
circuits can be characterized by the following equations:
Control Problem Formulation
Consider a general control system which consists of a plant
and a controller (Figure 1(c)). The plant (process) takes in
the input variable (d(t)) and control variable (CV) (u(t))
generating the process variable (PV) (y(t)). The controller
calculates an error (e(t)) signal as the difference between a
measured process variable and a desired setpoint (SP) (r(t)).
The controller aims at minimizing the error by adjusting
the process through the control variable (u(t)). The control
system can be characterized by the following equations.
ẋ(t) = f (x(t), u(t), d(t)), u̇(t) = h(e(t), u(t))
y(t) = g(x(t)),
e(t) = y(t) − r(t)
(21)
(22)
where f , g and h are arbitrary functions. Using a similar terminology, we can formulate the control problem for
the proposed system (Figure 1(c)). More precisely, the plant
(process) is described by the dynamics of the QS system of P.
aeruginosa, where the state variable X is the set of the concentration of chemical substances described by Eqs.(1)-(16)
(X = [C1 C2 C3 C4 C5 R1 R2 A1 A1EX A2 A2EX A1A A1AE ]
); the input (d(t)) can be viewed as the environment perturbations including bacterial growth, division, death, and
molecule fluctuations. The control variables (u(t)) are the
QSIs which target the LasR and AI. The genetic circuit can
be thought of as an integral controller which reacts to the
concentration of the LasR-AI complex.
To prevent the selective pressure caused by one specific
type of inhibitor, two types of inhibitors are generated simul-
2
10
1
10
0
Process Variable (y)
10
−1
10
−2
10
LasR Inhibitor (ρ = 0.9)
−3
10
AI Inhibitor (ρ = 0.9)
LasR and AI Inhibitor (ρ = 0.9)
−4
LasR Inhibitor (ρ = 0.7)
10
AI Inhibitor (ρ = 0.7)
LasR and AI Inhibitor (ρ = 0.7)
−5
10
−2
−1
10
0
10
10
Control Variable (u)
VE1 [E1EX ][A1EX ]
d[A1EX ]
= −d1 ([A1EX ]−[A1 ])−bA1EX [A1EX ]−
dt
KA1EX +[A1EX ]
(23)
d[A1AE ]
= −d3 ([A1AE ] − [A1A ]) − bA1AE [A1AE ] + k[E2EX ] (24)
dt
The process variable and reference signal is the concentration of the LasR-AI complex. We assume that the reference
signal (setpoint) is set below the activation threshold of the
autocatalytic process of the QS system. Therefore, the error
signal is approximately equal to the process variable.
There are two limitations of this scheme. First, given
any arbitrary genetic circuit, there is a maximum amount
of substances it can produce. If the maximum amounts of
inhibitors can not suppress the regulation system, then the
setpoint value is not achievable. Second, the biological parameters of genetic circuits put constraints on the realizability of the biological controller. To properly synthesize the
controller, we need to first examine whether the setpoint
is achievable. Once the setpoint is achieved, the minimum
amount of the inhibitors needed can be recognized as the
operation point for the biological controller.
4. CONTROL SYSTEM ANALYSIS
4.1 Control System Response
We first examine the open loop response without adding
QSIs. As shown in Figure 2, given a fixed extracellular
concentration of the AI, the response curve of the LasRAI concentration shows the bistability characteristics of the
QS regulation system of P. aeruginosa described by Eqs.(1)(9). When the extracellular concentration of the AI is below the activation threshold, the system maintains a relative
low concentration over all state variables. However, once the
concentration of the AI exceeds the activation threshold, the
concentration of all state variables changes the state from
low to high concentration sharply. This response is correlated well with experimental data in [21] which explains the
fact that when the population density of bacteria exceeds
the activation threshold (i.e. the accumulation of the AI),
it could induce virulence and biofilm formation.
The concentration of the LasR-AI complex in a low stable
state is different compared to the concentration in a high
(a) 30
25
Controller
20
15
0
10
(c)
10
5
0
0
LasR−AI (a.u.)
(b)
LasR
AI
Inhibitor (a.u.)
taneously and fed into the regulation system (plant) which
can be described by Eqs.(17)-(20). Hence, the control variables consist of four variables, namely, U = [E1 E1EX E2
E2EX ]. By adding kE2EX to Eq.(15) which accounts for the
catalysis of the LasR antagonist and replacing Enzyme in
Eq.(16) with E1EX , we form a closed-loop control system:
Figure 3: Dynamic range of the process variable (PV) with
different types of inhibitors where u represents concentration
of inhibitors and y is the concentration of the LasR-AI complex. The sharp decrease point of the PV can be recognized
as the activation threshold for the P. aeruginosa QS system.
LasR−AI (a.u.)
Figure 2: Bistablility of the P. aeruginosa QS regulation system. Given different cell densities (ρ), the steady state of the
LasR-AI complex concentration varies with different amounts
of fixed extracellular concentration of AI molecules.
0
10
500
1000
1500
2000
2500
3000
3500
Time(s)
4000
3000
3100
3200
Figure 4: System response by adding different QS inhibitors.
(a) Manually placing AI, LasR inhibitors and controllers (artificial cells) at different times. (b) and (c) represent the
concentration change of LasR-AI complex and inhibitors after placing the controllers.
stable state (Figure 2); this significant difference results in
the expression of the virulence factors. As shown in Figure 3, by varying the control variable (QSIs), the concentration of the process variable (LasR-AI complex) shows a
sharp decrease at a certain point which can be recognized
as the minimum amount of inhibitors that should be added
to the environment. Note that by using the LasR and AHL
inhibitors at the same time, the total amount of inhibitors
needed to repress the QS regulation system is minimized at
both cell densities (ρ = 0.7, 0.9 in Figure 3); this shows that
the multi-inhibitor strategy acts more effectively (the ratio
is 1:1 for both inhibitors).
By manually adding AI and LasR inhibitors at time 1000s
and 2000s, respectively, the concentration of the LasR-AI
complex decreases as expected. As inhibitors decays, the
system goes back to the high monostable state which can
induce virulence (Figure 4). Therefore, in order to keep
a low concentration of the LasR-AI complex, the continuous generation of inhibitors is necessary. At time 3000s, we
place the artificial cells into the environment to inhibit the
P. aeruginosa. The concentration of the LasR-AI decreases
and remains in the low stable state which proves that the
artificial cell can effectively control the concentration of the
LasR-AI complex.
4.2
Stability Analysis
The most important type of stability concerns solutions
near the equilibrium points. This can be examined by using
the theory of Lyapunov [15].
Definition 1. Consider nonlinear system ẋ = f (x), where
f : Rn → Rn (Rn is the set of real numbers); a point xe ∈ Rn
is an equilibrium point of the system if f (xe ) = 0.
Definition 2. A system is locally asymptotically stable
w/o Controller
(a)
0.9
(b)
30
w/ Controller
Monostable
High
Monostable
High
0.6
Promoter Strength (P)
Cell Density (ρ)
0.7
0.5 Bistable
0.4
0.3
0.1
Monostable
Low
15
ρ = 0.9
10
ρ = 0.7
0.2
0.3
0.4
0.5
1
2
Fixed Extracellular AI (a.u.)
0
200
3
4
5
Figure 5: Bistability w/o and w/ controller. The black and
white region correspond to monostable high and low, respectively. Without the controller, the gray region can be recognized as the bistable region.
(L.A.S.) near or at xe if there is an R > 0 s.t. |x(t = 0) −
xe | ≤ R ⇒ x(t) → xe as t → ∞.
In the case of biological systems, if some combinations
of state variables eventually converge to the same equilibrium point, then the union of all these combinations can
define an asymptotically stable region. On the other hand,
if some combinations of state variables can reach more than
one equilibrium point, then they define a multi-stable region.
As shown in Figure 5, for the P. aeruginosa QS regulation
system, there exist two monostable regions and one bistable
region. Given any set of initial conditions (here we only
consider varying the extracellular concentration of the AHL
molecules since the input signal mainly affects it) within the
monostable region, then the trajectories of state variables
will eventually converge to the equilibrium points. On the
contrary, the bistable region is not locally asymptotically
stable since it may reach low or high monostable states,
given different initial conditions.
QSIs generated by the genetic circuit are expected to increase the range of the monostable region since the inhibitors
can stabilize the system by repressing the concentration and
activity of both AHL molecules and the LasR protein. The
effects of the inhibitors are shown in Figure 5. As it can
be seen, the low monostable region in Figure 5(b) extends
to 10 times larger (the x-axis is extended) than in Figure 5(a) which confirms the functionality of the controller.
The bistable region disappears; only the monostable low and
high regions remain.
4.3
20
5
Monostable
0.2 Low
0.1
Cell Density (ρ = 0.9)
Cell Density (ρ = 0.7)
Cell Density (ρ = 0.5)
25
0.8
Sensitivity Analysis
Sensitivity analysis can provide valuable insights about
how robust the biological responses are with respect to the
uncertainty of biological parameters. It can also determine
which model inputs are the key factors that affect the model
outputs. By sweeping the model parameters compared to
their nominal values, it is possible to find sensitive parameters that can considerably change the model behavior (i.e.
the model is sensitive to those parameters).
However, some of the biological parameters may be hard
to realize due to the biological constraints [2]. Promoter
strength (PlasR ) and basal production rate (r) are suitable
for this design framework since we can tune their values
through the evolution method [2]. Given different cell densities (ρ), certain combinations of parameters (i.e. PlasR ,r)
can not repress the system (Figure 6). Hence, we should not
choose parameters within the colored region since they can
not repress the system toward the low monostable state.
ρ = 0.5
180
160
140
120
100
80
Basal Production Constant (r)
60
40
20
0
Figure 6: Sensitivity analysis for different combinations of
parameters of genetic circuit. The colored areas (red, green
and blue) indicate the high monostable state with different
cell densities. As cell density increases, the area expands.
(Note that blue and green regions are on top of red region)
5.
BIOLOGICAL PARAMETERS DESIGN
In this section, we provide a design procedure to design the
control circuit to produce the minimal amount of inhibitors
that can repress the QS regulation system. First, we need
to choose the operation points for the QS regulation system when the system is in high and low monostable states,
respectively. As shown in Figure 7, given an arbitrary cell
density, the minimum amount of inhibitors needed to repress
the regulation system should be recognized as the operation
point (i.e., points within the blue regions).
Next, by assuming that the overall system is at steady
state, we can set the derivatives of Eqs.(17)-(20) equal to
zero and solve for the steady state value of the control variable in high and low monostable states, respectively. Noticing that the concentration of LasR-AI (C1 ) is far less than
the disassociation constant (K) at low monostable state
while far larger at high monostable state, also, replacing
the concentration of extracellular inhibitors with the concentration of intracellular inhibitors, we can obtain:
Elow =
1
bEX + d
bEX + d
P , Ehigh =
P
bE bEX + (bE + 1)d r
bE bEX + (bE + 1)d
(25)
where Elow and Ehigh denote the minimum amount of inhibitors that should be expressed by the controller at low
and high monostable state, respectively. bE and bEX denotes the degradation constant of intracellular and extracellular inhibitors and d is the diffusion constant. P is the
promoter strength and r is the basal production rate.
We now give a concrete design example for a particular
environment. Suppose the input signal (d(t)) mainly results
from the variation in the number of the bacteria. Hence,
the cell density varies from time to time as a function of
the number of bacteria. We further assume that there is a
maximum number of bacteria that can be accommodated
in a specific volume due to the fact that bacteria can not
exceed the carrying capacity of the environment constraints.
Since the cell density can not exceed one, it is reasonable
to assume the maximum cell density is around 0.9. From
Figure 7, points A and B indicate the upper and lower value
needed to inhibit the regulation system. By using Eq.(25),
the minimum value of the promoter strength is 10 (a.u./s)
and the basal production rate r is 15. By checking the design
values in Figure 6, we can ensure this can repress the QS
regulation system.
0.45
4.5
0.4
4
0.35
3.5
0.3
3
0.25
2.5
0.2
2
0.15
0.1
*A
0.05
0
0
0.2
20
21.3924
18
21.3923
16
14
12
*B
10
1.08
6
1.07
1
4
1.06
0.5
2
4
SIMULATION
The stability and sensitivity analysis in previous sections
confirm the control feasibility by introducing artificial cells
as controllers. However, we only consider deterministic models in previous section. Variation in the QS systems for each
bacterium may result in different responses even with same
environment configurations. In this section, we use an agent
based simulator to validate the proposed system [23].
We assume that each bacterium can be modeled as a
sphere with radius 1µm which occupies approximately 4
µm3 . If the cell density ρ = 0.5, for a 3D lattice with length,
width, and height of 100 µm, then we need to put 1.25 × 105
bacteria to attain the required cell density.
As shown in Figure 8, the spatial and system responses
validate the bistability of LasR-AI and the effectiveness of
the genetic circuit controller. At t = 500s, the system is in
a high monostable state; we then place the controller into
environment. The controller first generates large amounts
of QSIs; after the system goes back into the low monostable state, the controller reduces the amount of generated
inhibitors to the basal level production. As the cell density increases, the transient expression of the inhibitors gets
higher due to the fact that the concentration of LasR-AI at
high monostable state is higher at high cell density. Additionally, the system response (which is similar to Figure 4)
validates our design.
7.
CONCLUSION
In this work, we have proposed an autonomous control
system that incorporates the QS regulation system of P.
aeruginosa and operates within an artificial cell. By analyzing system characteristics through numerical methods and
detailed simulations, we have shown that the artificial cells
can act as controllers able to keep the system in low monostable state; this prevents the expression of virulence and
biofilm development. We have also discussed general guidelines to synthesize such autonomous controllers in vitro; This
represents a first step towards a paradigm change in controlling the dynamics of the bacteria communication network.
8.
ACKNOWLEDGMENTS
This work was supported in part by the US National Science Foundation (NSF) under Grant CPS-1135850.
9.
x 10
8
2
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ρ = 0.9
ρ = 0.7
20
15
10
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21.3922
−5
Figure 7: Operational points for the QS system in low and
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6.
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