^{1}

^{3}

^{1}

^{2}

^{1}

^{3}

^{*}

Conceived and designed the experiments: QO YT. Performed the experiments: LJ YT. Analyzed the data: LJ QO YT. Wrote the paper: LJ QO YT.

The authors have declared that no competing interests exist.

_{d}_{d}_{C}_{C}_{R}_{d}_{C}_{R}_{c}_{R}

A computational model, based on a coarse-grained description of the cell's underlying chemotaxis signaling pathway dynamics, is used to study

Bacterial chemotaxis is one of the most studied model systems for two-component signal transduction in biology

(A) The

Significant progress has been made in several key areas towards quantitative understanding of the

As pioneered by Dennis Bray

Here, we briefly describe SPECS, a parsimonious model first introduced in

Following Tu et al.

The kinetics of the methylation level can be described by the dynamic equation

A simple phenomenological model is used here to model the

As first pointed out by Berg and Brown

For the boundary condition, we assume that when a cell swims to a wall, it swim along the wall for some time (1–5 sec.) before swimming away

The SPECS model was used to investigate

One central question in chemotactic motility is whether the drift velocity of the cells is determined by the gradient (

(A) Cell motion and intracellular dynamics in exponential ligand concentration profiles:

In exponential gradients, the average cell position increases linearly with time, leading to a constant chemotaxis drift velocity

The range of ligand concentrations over which the drift velocity remains constant in an exponential gradient is spanned by the two dissociation constants

The logarithmic sensing behavior, i.e., constant drift velocities in exponential gradients, predicted here from the model can be directly tested by measuring

Within the chemosensitivity range (

(A) The chemotaxis drift velocity

For

From Eq. 8, it is clear that

Taken together, a simple coherent picture of

In their early work, Berg and Brown first observed long tracks of

(A) Trajectory of a cell for 10 min for

The microscopic mechanism for the run length distributions can be studied with our model. We found that after some initial transient time, the kinase activity fluctuated around a constant average value

Analytically, the activity shift can be obtained by using the results from the last subsection and following the analysis in

In the natural environment, chemical signals not only vary in space, they also fluctuate in time. The fluctuation of a chemical signal (ligand concentration) sensed by a moving cell can be caused by: 1) randomness in the cell motion, i.e., the run-tumble motion and the rotational diffusion of the cell; and 2) temporal variation of the environment itself. Here, we investigated the effects of the latter due to ligand (spatial) gradients that also vary with time. In particular, based on the feasibility of future experimental tests of our predictions, we studied the case that

(A) Time dependence of the average positions of cells for three oscillatory gradients, all with the same amplitude but different frequencies (ω). The responses have the same frequencies as their driving signals, but the response amplitude decreases with the driving frequency. (B) The amplitude of the response decreases with frequency ω. The cross-over at low frequency (

How do the adaptation dynamics affect the cell's response to time-varying gradients? We investigated this question by varying

If the time-dependent term in the denominator of the integrant of the above equation is neglected for small amplitudes of cell motion, we can estimate the amplitude of cell motion at high frequencies:

The responses of

The capillary assay is an ingenious experimental method developed more than a century ago by W. Pfeffer and later perfected by J. Adler's group to study bacterial chemotaxis _{c} and C_{p} stand for the ligand concentrations in the capillary and the suspension pool respectively. _{0} and C_{1} respectively:

(A) Time-dependent ligand concentrations at three different positions in the suspension pool (see inset) from directly solving the ligand diffusion equation. The ligand concentration at a given position peaks at a given time, depending on its location.

From the spatial-temporal profile of the attractant, cell motion can be calculated by using SPECS. We considered the bacterial cells started randomly in a region of

Finally, we calculated the chemotactic responses in capillary assay for different values of

In this paper, a coherent picture of how

Calibrated quantitatively by the most up-to-date ^{3}–10^{4} cells in a linear ligand concentration profile and 10^{2}–10^{3} cells in a capillary assay

Perhaps equally important as predicting cellular behaviors, the SPECS model, which captures the essential features of the underlying pathway, enables us to understand these behaviors based on the key intracellular signaling dynamics, some of which can be difficult to study directly by experimental methods. For example, the constant drift velocity in an exponential ligand profile was found to be caused by a constant shift in the average kinase activity, which is maintained by a linearly increasing mean methylation level in balancing the exponentially increasing ligand concentration. At the individual cell level, this constant activity shift is also shown to be responsible for the intriguing observation that the average backward run time in an exponential gradient is similar to the average run time in the absence of a gradient, while the forward run time is much longer.

The SPECS model can be used to study various noise effects as well. The effect of the cell-to-cell variability for chemotaxis behavior in a linear gradient in a closed channel was studied by choosing (from a broad distribution) a random value for the internal parameters such as

The model framework described here lays the foundations for modeling cellular motility behavior based on the relevant underlying signaling pathway dynamics without describing the details of the individual signaling molecules. The current model can be extended in several directions to study many other interesting chemotaxis phenomena. For example, the interaction between the cells and the liquid-solid surface were oversimplified in this paper. Cells are known to turn with a preferred handedness when they run into a surface

Effects of CheY-p dephosphorylation and multiple motors.

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Chemotactic drift velocity and diffusion constant in exponential ligand concentration gradients.

(0.06 MB PDF)

Single cell behavior in different exponential gradients.

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Effects of cell-to-cell variability.

(0.06 MB PDF)