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Conceived and designed the experiments: JL TMM ZS. Performed the experiments: JL TMM ZS. Analyzed the data: JL JW ZS. Contributed reagents/materials/analysis tools: JL JW ZS. Wrote the paper: JL TMM ZS.

The authors have declared that no competing interests exist.

Chemotaxis is defined as a behavior involving organisms sensing attractants or repellents and leading towards or away from them. Therefore, it is possible to reengineer chemotaxis network to control the movement of bacteria to our advantage. Understanding the design principles of chemotaxis pathway is a prerequisite and an important topic in synthetic biology. Here, we provide guidelines for chemotaxis pathway design by employing control theory and reverse engineering concept on pathway dynamic design. We first analyzed the mathematical models for two most important kinds of _{1} (down cut-off frequency) and ω_{2} (up cut-off frequency) to optimize the chemotactic effect. Finally, we hypothesized a similar but simpler version of the dynamic pathway model based on the principles discovered and show that it leads to similar properties with native

The ability of motile microorganisms to respond to chemical gradients in their environment and direct their movement accordingly is defined as chemotaxis

Furthermore, we hypothesized a similar but simpler version of the chemotaxis dynamic pathway model by assembling the essential features of the native pathway together. From the view of control theory, chemotaxis pathway corresponds to the controller in bacterial moving system (

Chemotaxis pathway senses the ligand concentration and outputs the signal controlling motor bias in the form of [CheY-P]. Molecular motors on the cell surface can change cell's position or direction. In our research, the reverse engineering of the pathway is divided into three steps: analyzing the dynamic properties of native chemotaxis pathway; selecting and optimizing the transfer function describing the pathway dynamic; designing new pathway based on the transfer function. We changed the situation in our simulation by modifying the concentration field module in our program. The ligand concentration fields used here are mountain-shaped or peak-shaped.

In order to understand how chemotaxis pathway leads _{2}

Bode diagrams in

The figure shows the magnitudes and the phase shifts (with respect to the sinusoidal [L] signal) of [CheY-P] against different signal frequencies in non-adaptive chemotaxis pathway (A) and adaptive chemotaxis pathway (B) near typical steady states under four ligand concentrations: 0 µM,1 µM, 10 µM and 1mM, which can be further categorized into three working conditions (quiet condition: 0 µM; a fraction of receptors are occupied:1 µM and 10 µM; all receptors are saturated:1mM).

In order to test the chemotaxis effects of above two design strategies, we here apply two simple transfer functions to describe the dynamic relationships between receptor occupancy and [CheY-P] variance from a base line. Then we simulated the movement of bacteria under the control of these two types of filters. The base line of [CheY-P] is set to 2.71 µM, corresponding to 20% CW rotation bias. This makes the motility of unstimulated bacteria consistent to experiment

The influences of max amplification (A) and cut-off frequencies (ω_{0}, ω_{1}, ω_{2}) to chemotactic effect are tested in our simulation. The range of the cut-off frequencies used in the parameter scan is wide enough to cover both fast (e.g. phosphorylation) and slow (e.g. transcriptional regulation) biological processes. We simulated the movement of bacteria on petri disks, and found that the distribution of the bacteria is similar to the actual pattern obtained in experiments (simulated videos are available at

Obviously, the chemotaxis effects of the low-pass filter is determined by two parameters in the transfer function, A and ω_{0}, which correspond to the amplification and delay of the pathway, respectively. First, the max amplification A is positively related to the chemotaxis effects, if chemotaxis behavior is detected. A high amplification of the controller indicates that the pathway is sensitive to stimulation. The sensitivity should be strong enough to offset the tendency of stochastic movement, or otherwise, bacteria population would merely randomly swim in the median (_{0} should be high enough to enable the pathway to response to fast-changing ligand concentration. This transfer function is a typical inertia link. A low ω_{0} causes bacteria to respond to the input with a long delay. In this case if a bacterium is running in a wrong direction, it takes much time for the controller to transduce the signal to the motor and correct the cell; similarly if the direction of the bacterium movement happens to be correct, CCW bias of the motor cannot increase in time to keep the running state. Thus only the controller with very high max amplification and cut-off frequency shows significant chemotaxis behaviors (

One important feature inherent to this transfer function is non-adaptation, which means the steady state output of the pathway is related to the input signal level. Even though its amplification and cut-off frequency are both high enough, it still has some disadvantages. First, the mobility of the cell is high where [L] is high, which disables

First of all, we investigated how the three parameters in the transfer function (A, ω_{1} and ω_{2}) influence the chemotaxis effects. We simulated the movement of 100 bacteria and plotted the average [L] around each cell from 950 s to 1000 s to describe the chemotactic effect (

(A) Effects of variations of ω_{1}, ω_{2} on the average ligand concentration. The transfer function of the pathway is as shown in Eq. 2. The max amplification of the filter is fixed at 16. Each data point represent the movement of 100 bacteria in a mountain-shape concentration gradient (L = L_{0}exp(−x^{2}/r^{2}), L_{0} = 2 µM, r = 2 mm). The initial position of all the bacteria is (1.4 mm, 0 mm). 1000 s running was simulated and the average [L] in the last 50 s are calculated as a measurement of chemotactic effects. (B) Chemotaxis behaviors of cells guided by a band-pass filter with ω_{1} = 0.02 s^{−1}, ω_{2} = 5 s^{−1} in a mountain-shaped concentration field. The concentration field and initial point is the same as that in (A). Here, 1000 bacteria are simulated and their local [L] distribution is shown by box plots at each time point. Each box has three lines, which from low to high indicate the lower quartile, median, and upper quartile of [L] at a given time. Whiskers extend from the box out to the most extreme data value within 1.5 folds of the height of the box. [L] values beyond whiskers are marked by points. Solid line shows the average of [L] against time. The average ligand concentration increases fast and achieves a high final value (>1.9 µM) without oscillation. Few bacteria are trapped in places with low ligand concentration.

_{1}, ω_{2}. Generally, to a fixed down cut-off frequency ω_{1}, as up cut-off frequency ω_{2} increases from 0.3 s^{−1} to 100 s^{−1}, the final average concentration increases. But this contribution is not noticeable once ω_{2} is higher than 5 s^{−1}, resulting from the minimum response time limitation of the motor, which is set at 0.5 s in our model. Signals whose periods are much shorter than it can hardly influence the working state of the molecular motor.

Interestingly, the down cut-off frequency plays a critical role in adaptation via regulating the ‘memory’ of this pathway. Its best value is about 0.01 Hz or 0.02 Hz, which strikes a balance between sensing concentration ramps and adapting to long term stimulation, resulting in prompt approaching the concentration peak and maintaining its position (_{1} is too high, the ability of keeping the right direction is abolished, so the population average [L] increases relatively slow (_{1} is too low, when a bacterium runs over the concentration peak, it still “remembers” the former concentration rise and is misinformed to be swimming in the right direction so that cannot brake in time (

The controller described by Eq. 2 is perfectly adaptive, because its numerator is a differential link, which means calculating the differential of receptor occupancy. The sign of the outcome of the link indicates whether a bacterium is running up or down the concentration gradient. Compared to low-pass filter, the adaptation property of band-pass filter makes cell motility at steady state independent of input [L]. Thus, the distribution of bacteria around position with the highest [L] is more converged, and they would not be trapped at places with very low [L] (

We also investigated chemotaxis behavior controlled by band-pass filter in different conditions. The conditions we tested include placing bacteria at a position farther from the max ligand concentration (

It is interesting that all these simulations show very similar influences of magnitude and the two cut-off frequencies to chemotaxis effect (including higher rank band pass filters, provided the up and down cut-off frequencies are the same): high magnitude and up cut-off frequencies improve chemotactic result and the best down cut-off frequency is about 0.01 Hz or 0.02 Hz. Our finding here is possible to be an important guideline for chemotaxis pathway design. Better chemotactic ability, which could be recapitulated by the above-mentioned parameters, confers evolution advantage. Consequently, examining whether relevant properties of wild type

Based on the study on the kinetic features of the native chemotaxis pathway, we were able to design a novel pathway to control bacteria tropism by reversing the process from biological dynamics to transfer function. Traveling up the attractant concentration gradient is defined as positive chemotaxis, while the swimming away from repellants is called negative chemotaxis. We designed a pathway containing three molecules to realize a band-pass filter for positive chemotaxis. This pathway is proved to be the smallest structure that works as a band-pass filter (see

(A) Positive chemotaxis pathway. (B) Negative chemotaxis pathway. (C) Pseudochemotaxis pathway.

Evidently, the form of transfer function is made up of one differential link and two inertial links, very similar to Eq. 2. But because of nonlinear properties of biological systems, the magnitude and cut-off frequencies of the controller may vary under different [L] input. We keep the most important feature, down cut-off frequency d_{v} (molecule v deactivation rate) at the optimal value 0.02 s^{−1}, the magnitude (pathway sensitivity)as high as possible and up cut-off frequency (∼CheY-P dephosphorylation rate) high enough. In order to exclude any error stemming from the linear approximation, we tested the control abilities of the pathway designs and observed the chemotactic behaviors by our program based on the original differential equations (

Movement of 1000 bacteria is simulated in a mountain-shaped concentration field as the same in

The curve of the [L] against x is shown. (A) Simulated distribution of 2000 bacteria guided by the positive chemotaxis pathway. The initial position of the bacteria is (3.5 mm, 0 mm). (B) Simulated distribution of 2000 bacteria guided by the pseudochemotaxis pathway. The initial position of the bacteria is (1.4 mm, 0 mm). These two distribution patterns are very similar to actual experiment result

Some investigators consider the integral feedback as the key structure for robust perfect adaptation, which means if the activity of some molecules in the native chemotaxis pathway is changed, the pathway is still perfectly adaptive

Our work provides an example of applying control theory in dynamic model design (an important topic in synthetic biology). Bacterial chemotaxis pathway design mainly involves two aspects: one is the sensing of attractants or repellents by macromolecules and relaying these signals to the motor; the other is the quantitative relations between the input ligand concentration and output signal controlling motor rotation state. The second one is usually overlooked in previous studies, resulting in the previous pale imitation of the native chemotaxis process. Control theory is a powerful tool for designing system dynamics. The method used in our study, that is deriving transfer function from the mathematic model of a pathway, bridges the two realms. Linearizing the model describing a biological system is an approximate, which is only applicable to biological systems whose nonlinear effects are not very strong, but not to those working in saturation state or dead zone.

Transfer function and bode diagram are useful tools to describe the dynamics of a pathway in two aspects. First, they distill several “key parameters”, which are tightly related to the dynamic properties of the pathway, like A, ω_{0}, ω_{1} and ω_{2} in our research from large numbers of parameters in the pathway. This makes it much easier to scan the parameter space to determine the optimal value. And more importantly, different dynamic properties can be changed independently in this scanning. For instance, if we try to solely regulate the relaxation time of the pathway by changing ω_{1}, amplification or moving tendency of the cells will remain constant. Second, many properties of the bacterial chemotaxis are reflected in the transfer function. For example, by analyzing the transfer function, our research sheds light on the question of how

Reverse engineering is the process of discovering the technological principles of a device, object or system through analysis of its structure, function and operation by taking it apart, analyzing its workings in detail, and manufacturing a new one with similar function to the original

We need to know the transfer function of a biological controller in order to carry on reverse engineering. There are several approaches to uncover the frequency domain characteristics of the pathway and then derive its approximate transfer function. The one used here is calculating from the mathematical model based on detailed biochemical mechanisms, which relies on the existing knowledge of detailed pathway mechanisms. The second approach is to consider the controller need to be mimicked as a black box. We can input sine signals with different frequencies into it and analyze the magnitudes and phase shift of the output

The differential equation model for a signaling pathway can be written as:_{ij} = ∂F_{i}/∂x_{j}_{ij} = ∂F_{i}/∂u_{j}_{ij} = ∂G_{i}/∂x_{j}_{ij} = ∂G_{i}/∂u_{j}

The steady-state response of a system to sinusoidal input signal is defined as frequency-domain response. A signal can be decomposed into a set of sinusoidal signals by Fourier decomposition, thus the frequency domain properties reflect the dynamics of a system. If the system is linear, sinusoidal inputs always lead to sinusoidal outputs with the same frequency. The correlation between input and output can be characterized by two parameters, namely the Magnitude (the proportion by which the system amplifies the input sine wave) and the phase shift (the degree by which the output sine wave is delayed comparing to the input). To a linear system whose transfer function is H(s), its magnitude and phase shift to input with an angle frequency ω corresponds to the modulus and phase angle of the complex H(ωj)

Bode diagram shows the magnitude and phase shift of the frequency response of a linear system under different frequency. The magnitude is plotted in decibels (dB, computed as 20log_{10}|H(ωj)|), and the phase in degrees.

Our simulation program is made up of three modules: concentration field, motor and pathway (_{0}exp(−(x/r)^{2})). In other cases, a peak-shaped ([L] = L_{0}exp(−(x^{2}+y^{2})/r^{2})) field is employed. We apply a Markov chain with states 0 (tumble) and 1 (run) to represent the working state of molecular motor impelling

In each simulation, all the bacteria start at the same initiation position but their orientations are randomly selected. The initial concentrations of molecules in the pathway are set at equilibrium. In each time scale, if the motor state is 1, the cell runs forward at a speed 0.02 mm/s

To design the pathway systematically, we use a universal pathway model similar to the one described by Soyer O.S. et al _{L})), while last molecule is CheY and the concentration of its active form influences motor rotation bias. The biochemical dynamics for all concentration of activated molecules are described by following equations:_{it} is the total concentration of other molecules in the pathway except the receptor, and y_{i} is the concentration of activated molecule i. C_{ij} (D_{ij}) is the rate at which activated molecule j activates (deactivates) molecule i, C_{ii} (D_{ii}) is molecule i's self-activation (deactivation) rate, C_{i} (D_{i}) represents the rate at which the receptor activates (deactivates) the molecule i. Transfer function describing the correlation between u and [CheY-P] can be derived through the method described above (see

Supplemental materials. Exposition and derivation of transfer function of pathway near equilibrium and the model for molecular motor in

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Chemotaxis behaviors of cells guided by low-pass filter with various parameters (ω_{0}, A, whose values are shown above each subfigure) in a mountain-shaped concentration field. The concentration field is mountain-shaped (L = L_{0}exp(−x^{2}/r^{2}), L_{0} = 2 µM, r = 2 mm). In each subfigure, 100 bacteria start at (1.4 mm, 0 mm) and their movement in 1000s is recorded. The distribution of their local ligand concentrations is shown by box plots at each time point. Each box has three lines, which from low to high indicate the lower quartile, median, and upper quartile ligand concentration values ([L]) of bacterium population at a given time. Whiskers extend from the box out to the most extreme data value within 1.5 folds of the height of the box. [L] values beyond whiskers are marked by points. Solid line shows the average of [L] against time.

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Chemotaxis behaviors of cells guided by band-pass filter with various cut-off frequencies (ω_{1}, ω_{2}, whose values are shown above each subfigure) in a mountain-shaped concentration field. The max amplification of the filter is fixed at 16. The concentration field is mountain-shaped (L = L_{0}exp(−x^{2}/r^{2}), L_{0} = 2 µM, r = 2 mm). In each subfigure, 100 bacteria start at (1.4 mm, 0 mm) and their movement in 1000s is recorded. The distribution of their local ligand concentrations is shown by box plots at each time point. Each box has three lines, which from low to high indicate the lower quartile, median, and upper quartile ligand concentration values ([L]) of bacterium population at a given time. Whiskers extend from the box out to the most extreme data value within 1.5 folds of the height of the box. [L] values beyond whiskers are marked by points. Solid line shows the average of [L] against time.

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Effect of variations of ω_{1}, ω_{2} on the final average ligand concentration. Each data point represents the movement of 100 bacteria in 1000 s and their average ligand concentration in the last 50 s are calculated as a measurement of chemotactic effects. The transfer function, and max amplification (A), shape and parameters of the concentration field, and initial point of bacteria, are shown in

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CheY-P level changes caused by a step-wise signal in the band-pass filter. The parameters of the pathway are the same as that in _{2} = 0.16 s) and recovers to the basal level gradually (adaptation time τ_{1} = 40 s).

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Designed dynamics of positive, negative and pseudo- chemotaxis pathway and their block diagrams. (A) Positive chemotaxis pathway. Ligand quickly binds to molecule u to activate it. Activated u can dephosphorylate CheY-P and activate molecule v. Molecule v phosphorylates CheY-P. (B) Negative chemotaxis pathway. Ligand quickly binds to molecule u to activate it. Activated u can phosphorylate CheY-P and activate molecule v. Molecule v dephosphorylates CheY-P. (C) Pseudochemotaxis pathway. Ligand quickly binds to molecule u to activate it. Activated u can activate molecule v. Molecule v dephosphorylates CheY-P.

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Parameters of each subfigure in

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We are grateful to Dr. Chen GQ and Mr. Zou Y for enthusiastic comments on the bacterial chemotaxis pathway; to Drs. Li S, Zhao Q and Zhou T for enlightening discussions on control theory and providing us the cluster server. Furthermore, we would like to extend our thanks to the international Genetically Engineered Machine (iGEM) competition committee for providing us a platform to communicate with experts in this field.