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The authors have declared that no competing interests exist.

Bacterial chemotaxis is one of the most extensively studied adaptive responses in cells. Many bacteria are able to bias their apparently random motion to produce a drift in the direction of the increasing chemoattractant concentration. It has been recognized that the particular motility pattern employed by moving bacteria has a direct impact on the efficiency of chemotaxis. The linear theory of chemotaxis pioneered by de Gennes allows for calculation of the drift velocity in small gradients for bacteria with basic motility patterns. However, recent experimental data on several bacterial species highlighted the motility pattern where the almost straight runs of cells are interspersed with turning events leading to the reorientation of the cell swimming directions with two distinct angles following in strictly alternating order. In this manuscript we generalize the linear theory of chemotaxis to calculate the chemotactic drift speed for the motility pattern of bacteria with two turning angles. By using the experimental data on motility parameters of

Bacteria are the most numerous living organisms [

One of the most studied motility pattern is “run-and-tumble” of

Importantly, bacteria are able to alternate their motility pattern in response to gradients of certain signaling chemicals. Swimming cells sense the concentration of the signal and extend the duration of the runs, when moving in the direction of the chemical gradient. The chemotactic response of the cells is usually quantified by the average drift velocity in the direction of the gradient. After the key result of de Gennes, who proposed the so called linear theory of chemotaxis, the chemotactic drift speed was calculated for some basic motility patterns of bacteria [

Recently advances in bacteria tracking and a careful analysis of bacterial trajectories led to the discovery of more complex motility patterns. For example, a bacterium

In this manuscript, we provide an analytical calculation of the drift speed of chemotactic bacteria moving in a pattern with two alternating arbitrary turning angles. It is thus, to the best of our knowledge, the most general to date extension of the de Gennes result that can be applied to a broad class of bacterial motility patterns. Furthermore it allows us to predict the cell-to-cell variability in the drift speed of

In the following Section II we outline the derivation of the main result and in Section III combine it with experimental data on

Various bacteria utilize distinct swimming patterns to navigate their environment. Some of these patterns can be considered as two-step processes (“run-and-tumble” pattern of

Let the swimming pattern of bacteria consist of 4 phases: “Run 1”—movement along a certain direction, “_{1} with a corresponding average cosine value denoted by _{1}〉, “Run 2”—movement along the new direction, “_{2} with the average cosine denoted by _{1}〉. The speed of the bacterial movement between two subsequent rotations is considered to be constant and the same for both runs. Interestingly, there are reported cases when the forward and backward swimming speeds are different [

Despite the fact that the times at which the turning events occur are stochastic, the sequence at which the types of turning events follow each other is fixed. Note that, the pattern of _{run}. We should note that the run time distribution measured in experiments often deviates from exponential for short run times. In this respect the exponential distribution is a simplifying assumption, which is, however, crucial for analytical feasibility of the following calculations (for possible affects of non-exponential behavior see [

During every run the swimming direction of the cell fluctuates due to the thermal noise in the fluid and possibly due to active processes in the flagellar motor. This effect can be characterized by means of rotational diffusion with a constant _{r}. The value of _{r} can be measured experimentally and in general is in agreement with the estimate of Brownian rotational diffusion of a passive particle of the size of the cell [

A sketch of the trajectory of

During each run, the speed of the cell _{0} is constant. Motion is nearly straight and is affected by the rotational diffusion _{r}. The cell changes the direction of its motion during turning events (black dots), where turning angles Δ_{1,2} are allowed to have two different probability distributions. Importantly, the cell strictly alternates the two types of turning events. When swimming in the gradient of signaling chemicals ∇_{d} in the direction of the gradient, which we want to calculate.

In the presence of a chemical gradient bacteria are able to direct their overall random motion towards the attractant. Bacteria possess a chemo-sensory system allowing for temporal integration of the external chemical cues and a delayed response which biases the rotational direction of flagellar motors. When the cell climbs up the gradient it can extend durations of the run phases. The biochemical structure of the chemotaxis response is well understood at least for _{0} = 1/_{run} and _{d} in the concentration of the chemoattractant with a small linear gradient pointing along _{β} > 0. Then we can determine the drift velocity as the sum of average displacements of a run _{run}:
_{β} on a particular path. Obviously the drift velocity of the bacterial population with

As was shown by de Gennes, we can first analyze the drift velocity _{δ} for a simplified response kernel:
_{δ} is the chemotactic drift speed for the delta-response

The derivation of de Gennes [_{z}(_{z}(_{δ}, _{δ}, _{δ}:
_{αβ} = _{αβ} = −1 + _{j}(_{j}(_{r} and λ_{0} follows that the drift velocity is always inversely proportional to the base-line turning rate λ_{0} and to rotational diffusion coefficient in the third degree, i.e. _{0}. This scaling can be qualitatively understood as follows. The length of each run of the cell is proportional to its speed, while the bias in this run is determined by the sensed gradient. During a run, the cell translates the spatial gradient into the temporal concentration gradient where the cell velocity enters as a scaling factor, thus resulting in the overall quadratic dependence of the drift speed on cell velocity. Before applying the general result of

The drift speed is linearly proportional to the gradient of concentration |∇

Due to the specific form of the response function,

We model the motility of an ensembel of ^{6} cells during the time interval _{0} = 45 ^{−1} and _{run} = 0.3 s [_{r}, as well as parameters

For _{r} = 0 rad^{2}s^{−1}, analytically obtained drift velocity in the general case of two alternating turning events is presented as a green surface in _{d}(_{d} = 0. Agreement is similarly good for the case _{r} = 0.2 rad^{2}s^{−1}, see results on

(a) Analytically obtained function _{d}(_{r} = 0.0 rad^{2}s^{−1}, |∇^{−4} and ^{3}. (b) Comparison of analytically (lines) and numerically (symbols) obtained drift speed dependences on the parameter

(a) Analytically obtained function _{d}(_{r} = 0.2 rad^{2}s^{−1}, |∇^{−4} and ^{3}. (b) Comparison of analytically (lines) and numerically (symbols) obtained drift speed dependences on the parameter

Numerical experiments are consistent with analytical calculations within 10% error up to the gradient value |∇^{−4},

Analytically obtained predictions (curve) agree with numerical results (symbols) up to the gradient values of ⋍ 0.5 ^{−4} (_{r} = 0.2 rad^{2}s^{−1}, λ = 3.3 s^{−1},

Recently, the trajectories obtained by the high-throughput 3D bacterial tracking method [

(a) Histograms showing the distribution of flick angles with various body lengths (data from [^{−4}, λ = 3.3 s^{−1}, _{0} = 45 ^{−1}, _{r} = 0.2 rad^{2}s^{−1}, ^{3} and

Importantly, comparing the measured angles for the cells with different sizes revealed that individuals actually show very narrow flick angle distributions with different means [orange, yellow, green and cyan histograms in

As was shown in [

Therefore, given the more detailed data of [^{−4}, λ = 3.3 s^{−1}, _{0} = 45 ^{−1}, _{r} = 0.2 rad^{2}s^{−1}, ^{3} and

Continuously advancing measurements techniques allow us to get a more detailed information on a bacterial behavior at the level of individual cells. We are at the point when the variability between cells can and should be accounted for in our quantitative analysis of motility and chemotaxis. In this work we considered one of the most general swimming patterns containing two alternating turning angles. This analytical framework allows for a straightforward analysis of

Although we observe a noticeable difference in the drift speeds of bacteria of different sizes, for the considered small gradients this difference is of the order of 10%, see

The curves represent the angle-dependent characteristics obtained from Eqs ^{−1}, _{0} = 45 ^{−1}, _{r} = 0.2 rad^{2}s^{−1} and

Another important parameter affecting the drift speed is the rotational diffusion (cf. Figs

For bacterial chemotaxis it is not only important how fast cells can move towards the higher concentration of signaling chemicals, but also how well can they localize themselves near the source of the gradient. Thus not only the drift velocity but also the effective diffusion constant of the bacterial motility pattern play an important role [_{d}/

As for the drift velocity, substituting into _{0}. This scaling is due to a simple random walk estimate of the diffusion constant as the mean squared step distance (_{0}_{run})^{2} divided by the mean run time _{run}.

With this information at hand we have a full theoretical tool set to quantify bacterial chemotaxis in the small gradient approximation. The theoretical results presented here for a 4-step pattern, allow us to predict the drift as a function of the cell size. Importantly, these predictions could be verified experimentally where the drift speeds of cells along the linear gradient can be correlated with their size. One of the possible applications of the size-dependent drift velocity is in cell sorting, where after a certain time of motion along the linear gradient the cells of different sizes would be found at distinct positions corresponding to their drift speed.

We think that the analytical relations, such as chemotactic drift speed considered here, provide a rigorous link between motility pattern and chemotactic response and thus can be used in experiments to infer the yet unknown parameters of bacterial sensitizing based on tracking or chemotactic drift experiments.

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