Designing synthetic biological circuits using genetic elements
has become a research field for developing novel biochemical sensors.
Specifically Bacterial cell based biosensors have been studied for the environmental
monitoring, clinical diagnostics and drug discovery. One of the main challenges
for this biosensors to meet real-life applications is the low sensibility and
deficient detection limits.
The rise of multiple antibotic resistant bacteria and the evident crisis
in drug discovery has become one of the main challenges in human history.
Traditional screening not only takes longer times,from years to decades,
but often fails to discover novel biomolecules. Bacterial biosensors have provided
a novel benchmark for the screening of secondary metabolites producers, boosting
the discovery of antibiotics and broadening the spectrum of the latter.
One of main aspects that should be considered for the development of novel
antibiotic biosensors is the sensibility, since they can only work properly below
the minimum inhibitory concentration.
Our project consists of a series of engineered multi-layered transcriptional amplifiers that
sequentially increases the output expression level of a GFP reporter protein by the presence of an
antibiotic belonging to a specific mechanism of action (MOA) in order to enhanced the process of drug
discovery and bacterial screening.
Modelling Biological systems has become an area of interest for multidisciplinary areas of science,
medicine and engineering. Mathematical models allow to understand and predict the behavior of complex
systems using simple concepts.
Bacterial bio-sensors can be divided into three modules, the first comprising a sensing module that
recognizes the external signal and transduces into a transcriptional output, the computing module that
modulates the transduced sensor signal and the output module which executes the physiological output
response. In this project, we develop mathematical models for the engineered multi-layered
transcriptional amplifiers which acts as a computing module using both exact deterministic differential
equations and stochastic simulations to provide a proof of concept for the amplification signal of the
antibiotic biosensor.
The structure of the following sections goes as follows: First we derive the mathematical equations for
an already characterized heavy metal biosensor consisting of a constitutive promoter expressing a
repressor protein inhibiting the later expression of a reporter gene. Then we will consider the effect
of adding a mono-layer orthogonal transcriptional amplifier to the signal amplification. After that we
will extrapolate this same models to our antibiotic biosensor.
Figure 1: Differential equations of whole system without
cascade amplification signal . The constants where chosen
arbitrary taking the following values: alpha0 = 0.05`, beta0
= 0, beta1 = 0.015`, deltaf = 0.005`, deltaR = 0.025`, delta
= 0.00025`, k1 = 0.04`, k2 = 0.80`, kmin1 = 0.015`, kmin2 =
0.075`, P(0) = 1, R(0) = 0.05, RP(0) = 1 GFP(0) = 0.`, I(0)
= 2, IR(0) = 0`}
The fisrt system to be modelled consist of an heavy metal
biosensor. In the abscence of heavy metal the constitutive
promoter expresses a repressor protein that inactivates the
expression of gfp. Some of the requirements for this system
to work correctly is a low efficiency constitutive promoter
such that in the presence of heavy metal inducer the gfp
signal can be turned on. Another important requirement is
that the arsenic repressor interactions be stronger than the
promoter-repressor in order to overcome such inhibition and
lower the detection limits of the heavy metal.
The system of differential equations of this system is:
Consider the first system of differential equation. Notice
that in chemicalequilibrium of the the rate of
Repressor-Promoter formation is equal to therate of
dissociation of the later. This processes of binding are
know to hap-pen at short life times so that an equilibrium
between Repressor-Promoteris stablished so quickly so that
their concentrations dont change over time. Therefore a
steady state approximation can be proposed. The chemical
equi-librum constant k−1k1=RPC1=Keq1 is obteined directly
from the steady state assumingd C1(t)dt= 0. Therefore in
chemical equilibrium the terms for pro-moter ocuppancy
vanish and the differential equation simplifes to:
We can repeat the same process for the formation of
inhibitor repressorcomplex, assuming a steady state for the
binding of Inducer to the RepresorProtein. SettingdC2(t)dt=
0. We obtain a chemical equilibrium constantdirectly from
this assumptionk−2k2=IRC2=Keq2, Once again some terms
arecancelled out and the system of differential equations
simplifies to:
The analytical solutions of R(t) and I(t) areR(t)
=αδR(1−eδRt) +R0eδRtandI(t) =I0eδIt.In steady state R
becomes the rate of basal expression anddegradation dilution
and I becomes zeroRss=αδR. Assuming that the rateof basal
expression is approximately zero the last equation becomes
A simple empirical equation can be obtained by setting the
promoterocuppancy as and changing P(t) to the promoter
derepression asO(R) =P(t)P(t)+C1(t)=Keq1R(t)+Keq1However we
are interested on the effect of the in-ducer in the
expression of GFP therefore we can change the promoter
de-repression(induction) asO(I) =I(t)I(t)+KFor experimental
purposes I can be considered constant at all time, thatis,
the degradation rate of I is small so thatdI(t)dt=
0setingδI= 0. Theexpression of can be adjusted to a
Hill-Langmuir fucntionP(t)P(t)+C1(t)=InIn+Kn
Multi-layer orthogonal transcriptional amplifier have proved to amplified considerably the signal
output lowering the detection limits for sensing applications. The system of differential equations
for a monolayer- orthogonal transcriptional amplifier consisting of a dimer activator previously
studied integrates the following terms.
For a bilayer biosensor the mathematical approach for stablishing the differential equations is the
same
Instead on trying to find the analytical solutions of the systems of differential equatios. We are
now interested in compairing the biosensor with and without a multilayer- orthogonal transcriptional
amplifier under same circumstances.
Microfluidics Model
Compared to single phase flows, microfluidic two-phase flows relies on several physical phenomena that
need to be controlled for the droplets to fully form. The flow properties in microchannels rely on three
parameters: the channel geometry, the properties of both fluids, and the flow conditions. These factors
can be described by some important dimensionless parameters.
The physics behind microfluidics include the relations between the interfacial tension, inertial
forces and the involved fluids’ properties. Dimensionless numbers have become a standard manner to
compare fluid interaction at macro and micro scale.
The flow in microfluidic systems are usually characterized by low Reynolds number values, which
describes the ratio between inertial and viscous forces in fluids, and can be used to characterize the
system.
EQUATION 1.1
Where ρ is the density of the fluid (kg/m3), V is the average fluid velocity (m/s), L is the linear
dimension (m) and µ is the dynamic viscosity of the fluid (kg/ms). In microfluidic systems, viscous
forces (µ) dominate and Reynolds numbers are generally smaller than 100, leading to the prevalence of
laminar flow. For Re << 1 the flow is dominated by viscous stresses and pressure gradients, hence
inertial effects are negligible, and the trajectories of fluidic psections can be controlled
precisely.
The dominant forces at the microscale are interfacial and viscous forces, therefore it is
important to determine the relative importance of the interfacial tension compared to other forces
in droplet generation. The capillary number Ca is the ratio of viscous stress to capillary pres
EQUATION 1.2
Here η is the viscosity of the fluid in the two-phase system, μ is the velocity of the phase, and Υ is
the interfacial tension of the liquid-liquid interface. At low Ca (<1) the interfacial tension
dominates, and spherical droplets are found. In contrast, at high Ca (>>1) the viscous forces play
an important role, leading to deformation of the droplets and sometimes to asymmetric shapes.
Suryo & Basaran had previously reported a phases map in which different droplet generation regimes
were identified at different ranges of Ca numbers for both continuous and disperse phases [numero].
The squeezing regime generates well-rounded monodisperse droplets at high throughput and has defined
boundary conditions for the Ca numbers as shown in Figure 1. Hence a prediction can be made based on
the parameters considered in the Ca in order to generate monodisperse droplets in the squeezing
Figure 1. Phase diagram in (Cac,Cad) plane for various modes observed in microcapillary devices
The interfacial effects become relevant when working at microscale and are crucial in multi-phase
flows. The interfaces considered in microfluidic two-phase systems include the fluid-wall and
fluid-fluid interfaces. The wetting properties of the fluid-wall interface are important to determine
whether there will be an ordered droplet production or not. If there is a complete wetting of the
continuous phase in the microchannels, an orderly pattern can be achieved. The hydrophobicity or
hydrophilicity of a solid surface can be expressed quantitatively by contact angles. The contact angle
between a liquid and a solid is the angle formed by the tangent from the contact point along the
gas-liquid interface. If the contact angle between a liquid and a solid is less than 90° the liquid
will wet the surface and spread over it. If the contact angle is ≥90°, the liquid will stay on the
surface as a bead. Therefore, the* contact angle between a liquid and a solid is dependent on the
nature of the liquid as well as the surface characteristics of the solid. Water-in-oil (W/O) droplets
can be achieved in hydrophobic surfaces that have a contact angle higher than 90°, in which the
droplet will not spread over the surface; nevertheless, it is possible to modify the contact angle by
adding surfactants at different con
Surfactants are often used in order to modify the contact angle between the fluid-wall
interface. They are also used to reduce the interfacial tension between fluids and to prevent
the coalescence or merging of droplets [13]. In the presence of surfactants, the interfacial
tension is determined by the competition between interfacial deformation and surfactant
convection, diffusion and adsorption–desorption kinetics during droplet generation [3]. A faster
interfacial deformation and slower mass transfer process causes the surface coverage of
surfactants to become smaller, thereby the interfacial tension turns larger [13].
Properties of the fluids (i.e., viscosity, density, interfacial tension) and design parameters
(i.e., geometry, dimensions, flow rates) are the main variables responsible for the formation of
continuous monodisperse droplets in a microfluidic system.
The droplet breakup process in microfluidic devices has been extensively studied, and several
mechanisms have been classified based on the droplet generation. The most important mechanism is
the squeezing regime, since this produces continuous monodisperse droplets in defined intervals
of time.
The droplet breaking process is not fully understood since it is dependent on many parameters,
nevertheless a good approximation can be made based on design parameters including flow rate,
viscosities, and geometry of the device.
As the viscous forces and capillary forces are the dominant forces in the droplet breakup and
capillary numbers relate those terms, they are essential in the determination of design
parameters for droplet formation. Based on the known physical parameters of the fluids being
used, it can be possible to calculate the best velocities at which the fluids begin the droplet
formation in the squeezing regime.
A MATLAB script was developed in order to determine the optimized velocities and hence the flow
rates for droplet generation based on the boundary conditions of the squeezing regime. The
following figure is an example of the scatter map created based on the conditions available in
literature, the physical parameters of the fluids (interfacial tension, density, and kinematic
viscosity), and the geometry of the channel (width and length). The flow rate conditions were
used as parameters into a COMSOL simulation in order to determine the optimized flow rates that
could generate continuous monodisperse droplets in order to try them in the experimental section
of the microfluidic device.
As previously discussed, the role of viscosity, effect of flow rates, and geometries of the
microfluidic device are of great relevance on droplet formation [1]. Nevertheless, the experimental
and empirical investigations that focus on these effects are usually subject to spend a large amount
of resources, time, and effort in order to achieve an optimized set of conditions for droplet
generation, since there is a high chance of failure at initial stages of the design process [2],
[3]. Therefore, a numerical study of microdroplet generation could provide a suitable model for the
prediction of the effects of the previously mentioned parameters on a droplet generation T-junction
microfluidic device.
A reliable simulation may be a suitable method to reduce the required time to achieve an optimized
characteristic of the system, and to be able to forecast how the different modifications of the
parameters will impact on the droplet generation [3]. In order to determine the effect of different
flow rates of both phases and their relation to droplet size and monodispersity, simulations were
carried out using COMSOL Multiphysics which includes a computational fluid dynamics (CFD) module and
a microfluidics module as well [4].
CFD provides a reliable alternative in order to obtain insights into a complicated process. Several
methods have been typically used in order to simulate two-phase fluidic systems, including volume of
fluid (VOF) method, level-set method (LS), phase-field method, and lattice-Boltzmann method.
Although there are several advantages regarding the different methods, the LS method represents the
interface by a smooth function, and it is convenient for calculating the curvature and surface
tension forces [5]. Hence, it seems to be suitable for modeling droplet breakup process in
microfluidic devices. In the proposed simulations, we employed LS method to study the
droplet-breakup process using FC40 as the continuous phase, and water as the dispersed phase. The
effects of the capillary number (Ca) and the different flow rates are investigated [3].
COMSOL Multiphysics determines the modules that are needed in order to give solving parameters, the
dimensions in which the experiment will be performed, and the definition of the physics of the
problem. For this simulation, we used a laminar two-phase flow using the level set method. A
single-phase flow system is less complicated to model computationally than the multiphase flow due
to fewer partial differential equations that need to be solved parallelly. Nevertheless, it is
needed a two-phase system in order to properly simulate the relationship between the dispersed and
continuous phases.
In order to analyze the motion of a liquid, the starting equation to use is the Navier-Stokes
equation. The following assumptions are made in order to simplify the model: a constant fluid
density, a laminar flow regime exists throughout the system, all fluids are Newtonian, and
three-dimensional stresses for a fluid obey Hooke’s law [6]. While assuming the above, the
incompressible form of the Navier-Stokes equation is as follows:
The first design is a three-dimensional T-junction composed of two inlets. The inlet for Fluid 1 had
a width of 100 µm and a length of 400 µm prior to the junction. The inlet for Fluid 2 had an inlet
width of 100 µm and length of 300 µm prior to reaching the junction, as can be seen in the following
figure. Post T-junction, the droplets travel 600 µm to the expanded pillar induced merging chamber.
The entire geometry has a depth of 100µm.
A free tetrahedral mesh with a COMSOL Multiphysics® predetermined element size of “fine” was
utilized. Then a mapped operation was performed over the different distribution elements of the
geometry in order to have smaller features in the interface section. Finally, a swept function was
used to generate the different regions of the mesh size in the model.
The properties of fluids utilized in the three-dimensional study can be seen below in Table I. The
flow rate of the continuous phase is not constant but varies from 1µl/min to 50µL/min. The
variables for the oil are based on the specification for FC40.
Table 1. Properties of fluids used in three-dimensional study.
The main goal of the simulation is to generate droplets. Droplet generation commonly occurs by
shearing one fluid phase with another. Utilizing the T-junction defined previously with the
parameters based on the MATLAB script. All physical parameters are based on literature. Since the
goal was to see whether the conditions could generate droplets or not, the results are only
qualitative with a binary response on whether or not a droplet was formed. The next steps for this
simulations were to test the given parameters that could make droplets in the experimental setup.
REFERENCES
Assumptions for simulations
A constant fluid density
A laminar flow regime exists throughout the system
All liquids are Newtonian fluid
Assume the three-dimensional stresses for a fluid obey Hooke’s law
Taking this into account we run the experiments in order to generate droplets, we can see the
comparison between the simulation where droplets were generated and the real experiments using the
flow rates that the simulation results suggested.
