• Introduction
• Defining The Best Topology
• Sensitivity analysis

Band-pass Filter

Introduction

Here we focus on developing advanced equipment for our toolkit to implement fast, economical and convenient measurement of aromatic compounds in environment.

Common reporting systems failed to meet this requirement because they often possess a Hill-function type dose-response curve. As can be observed from the graph of a typical Hill function (Fig.1, a), the linear range of a Hill function could be rather narrow, and the transition from low-output to high-output may be quite obscure to naked eyes. Thus appropriate equipment would be required to accurately measure output that follows Hill function type dose-response curve, making the measurement expensive and time consuming.

Figure 1. Dose-response curves and typical measurement results for a canonical reporting system a, and band-pass filter b. a, A general reporting systems typically possesses a Hill function type dose-response curve, and it's quite difficult to determine the absolute intensity of a particular signal among its gradually increasing outputs. b, dose-response curve of a band-pass filter possesses a single peak, and it's relatively easy to determine the position of the peak in its output series.

Although unaided eyes can barely determine the absolute intensity value of a particular signal among a series of signals with various intensities (Fig.1, a), humans are pretty competent at determining which signal is the strongest one, especially when there is a single peak among the signals (Fig.1 b). Thus it can be reasoned that if we are capable of transforming a series of signals with intensities changing monotonously into a series of signals with an unique intensity peak, reading and interpreting of the output signals will become much more intuitive and convenient. Fortunately, a band-pass filter is exactly the equipment that can turn a series of gradually increasing input signals into a series of output signals with a single peak.

It can be expected that when a band-pass filter is successful constructed, we may serially dilute our sample into a concentration gradient and put our biosensor into the sample. The analyte concentration can be easily determined by the serial number of the test tube exhibiting highest output intensity (Fig.2). We hoped that by implementing a band-pass filter circuit in our bacterial host cells, we might realize fast, economical and convenient detection of aromatic compounds in environment.

Figure 2. Graph illustration of proposed band-pass filter testing method. First a sample series need to be created by serially diluting the original sample. Then bacterial cells expressing band-pass filter circuits will be exposed to the sample series and the concentration of original sample will be determined based on serial number of the sample inducing highest output.

The Concept of Band-Pass Filter

Band-pass filter is a term used in electric engineering. It describes a device that passes signals with frequencies confined to a certain range and blocks signals with frequencies outside that range. The band-pass filter is constructed by combining a high-pass filter, which only pass signals with high frequencies, and a low-pass filter, which only pass signals with low frequencies (Fig.3).

Figure 3.Sketch diagram of a typical band-pass filter in electric engineering. Vertical arrows show the input-output relationship of individual high-pass filters (left circle), and individual low-pass filters (right circle). The horizontal arrows show input-output relationship of a band-pass filter constructed by concatenating a high-pass filter and a low-pass filter. In an electric band-pass filter, the input signal is first processed by the high-pass filter to filter out low-freqeuncy signals and then processed by the low-pass filter to filter out high-frequency signals, leaving only medium-frequency signals.

In analogy to an electric band-pass filter, a biological band-pass filter is a device that can be activated only by an input signal with medium intensity. Neither signal with low nor high intensity will generate an output signal (Fig.4).

Quite similar to an electric band-pass filter, a biological band-pass filter can also be separated into two components, namely the two types of regulation the input node exerts on the output node in the network topology. In one way, the input node activates the output node through a positive feed-forward loop; in another way, the input node inhibits the output node through a negative feed-forward loop. Such a network topology, with two counteracting regulating feed-forward loops connecting input node and output node, is called an incoherent feed-forward loop topology (Fig.4 a). The positive feed-forward loop will respond only to high intensity input signal (Fig .4 b),serving as a 'high-pass filter'. The negative feed-forward loop will respond only to the low intensity input signal (Fig.4 c), serving as the 'low-pass filter'. By fine-tuning transition points of the dose-response curve of the incoherent feed-forward loops so that the transition point of the negative loop is higher than that of the positive loop, the biological band-pass filter, constructed by combining these two loops together, will respond only to a medium intensity input signal and generate an output peak at a specific concentration (Fig.4 d).

Figure 4. Sketch diagram of a possible topology (a) and functioning mechanism (b, c and d) of a biological band-pass filter. a, A network topology containing an Incoherent feed-forward loop, capable of generating a band-pass filter. The input node A directly represses output node C, creating a negative feed-forward loop, while indirectly activating output node C through repressing internode B which represses node C, creating a positive feed-forward loop. b, dose-response curve of positive feed-forward loop when characterized independently. The positive loop will respond only to high intensity input. c, Dose-response curve of negative feed-forward loop when characterized independently. The negative loop will respond only to low intensity input. (d)The integrated dose-response curve of the incoherent feed-forward loop. High intensity input is filtered out by negative loop and low intensity input is filtered out by positive loop, only medium intensity input will induce a significant response.

Constructing Band-Pass Filter

To rationally design the circuit for our band-pass filter, we first selected three potential circuit networks containing a core topology: incoherent feed-forward loop.

We then used Ordinary Differential Equations (ODEs) to analyze these circuit networks to identify the most robust circuit network and finally chose to follow a four-node network.

To select appropriate proteins to function as individual nodes in the circuit network, we first figured out the most crucial parameters through a parameter sensitivity analysis and determined the most desirable value for these parameters. Then we chose regulatory proteins whose kinetic parameter values are close to the desirable values, reasoning that they would work much more efficiently than casually chosen ones.

Based on the analysis above, we selected phage transcription activator phiR73delta as internode activator and phage transcription inhibitor cI as internode inhibitor while choosing NahR as the input sensor and sfGFP as reporter. The final construct is shown in Fig Details of constructing process can be view at Model.

Construction of our hybrid promoter

After deciding the circuit and protein to use, we still need to address an important issue: we need to find a way to enable synergic regulation of the reporter gene by two different transcription regulators. So we modified bacteriophage φR73’s promoter into a hybrid promoter that can be activated by the PhiR73 delta activator and repressed by the repressor cI simultaneously and put reporter sfGFP under its regulation. We replaced the sequence between the -10 and -35 elements of the promoter with the cI binding site Or1 from Phage λ. When PhiR73δ activator binds to its binding site upstream of -35 element, the transcription will start. The binding of cI dimers will block the binding of σ70 factors and thus repress the transcription (Fig.6).

Figure 5.The Construction of Our Hybrid Promoter.

As the input signal increase, the promoter will function in the following way: when the input signal is weak, the concentration of PhiR73 delta is too low to generate a strong output; when the input signal is medium, despite a portion of promoters occupied by cI dimmers, the rest still can be activated by PhiR73 delta and bring about a visible output; when the input signal is strong, almost all of the promoters are blocked by cI dimers and the output is shut down. Hence only medium input signal can generate a significant output and the Band Pass Filter was made.