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Implications of 3-step swimming patterns in bacterial chemotaxis

We recently found that marine bacteria Vibrio alginolyticus execute a cyclic 3-step (run- reverse-flick) motility pattern that is distinctively different from the 2-step (run-tumble) pattern of Escherichia coli. How this novel swimming pattern is regulated by cells of V. alginolyticus is not currently known, but its significance for bacterial chemotaxis is self- evident and will be delineated herein. Using an approach introduced by de Gennes, we calculated the migration speed of a cell executing the 3-step pattern in a linear chemical gradient, and found that a biphasic chemotactic response arises naturally. The implication of such a response for the cells to adapt to ocean environments and its possible connection to E. coli 's response are also discussed.

Existing observations made in Escherichia coli (E. coli ) have shown that sensing and motility impose different requirements on bacterial chemotactic response (1,2). The debate on this interesting issue was initiated by the observation of Block,Segall,and Berg (3) who discovered that the experimentally measured chemotactic response function R(t) integrated over time t is zero. In physical terms R(t) can be thought as the Green's function of the chemotactic network when subjected to an impulsive or a δ-in-time perturbation. The importance of this null integrated effect goes without saying, and was immediately recognized by the investigators as the bacterium's means of sensing. In their words (3), "the bacterium compares the information received in the past one second with that received over the previous three seconds." In effect, the double-lobe response function, which is displayed in Fig. 1(c), allows the bacterium to react to fast temporal variations of a chemical signal c(t) but not to its dc component, enabling the cell to adapt to a wide range of chemical concentrations. Using a macroscopic diffusion argument, it was suggested by Schnitzer et al. (4) that a finite memory time is required for a bacterium to migrate in a linear chemical gradient; without the memory effect (or R(t) ≃ δ(t)), it was concluded that the chemotactic coefficient κ = V /∇c or the drift velocity V would be zero, where V is in the direction of the chemical gradient ∇c. However, de Gennes pointed out that the macroscopic diffusion approach ignored important correlations between bacterial swimming and the underlying chemical gradient (5). By taking into account such correlations, de Gennes showed that the optimal (or a fast) response for migration in a linear gradient is an exponential function with a decay rate determined by the cell's memory time τ . He further pointed out that the double-lobe response function observed in E. coli could only reduce the migration speed in the gradient.
Contributing to this stimulating debate is the finding of Clark and Grant (1), who argued that while a cell needs a fast drift speed in a concentration gradient, it is equally important for the cell to localize once the top of the gradient is reached. They showed that the singlelobe function proposed by de Gennes is inadequate for cell localization. By imposing the co-requirements of being able to localize as well as to migrate, they demonstrated that the optimal response function is biphasic, which is in remarkably good agreement with the one measured in the experiment (3). This observation led Clark and Grant to conclude that the biphasic response in E. coli perhaps reflects a compromised need of the cells in different environments. A recent study also suggested that the laboratory observed bacterial response corresponds to the maximin strategy that ensures the highest minimum uptake of nutrient for any profile of concentration (2).
Recently, we found that the swimming pattern of the marine bacterium V. alginolyticus is a cyclic 3-step process (8), where a cell swims forward for a time interval ∆ f and it then backtracks by reversing the motor direction for a time ∆ b . However, upon resuming forward swimming, the bacterial flagellum flicks causing the cell body to veer in a new direction. This type of motility pattern is very different from that of E. coli, which exhibit a runtumble pattern. By way of introduction a typical trajectory of V. alginolyticus and of E. coli are presented respectively in Fig. 1(a) and (b). For the V. alginolyticus' trajectory, Fig.  1(a), the forward and the backward segments are designated by green and red, respectively, for clarity. We termed this novel swimming pattern a run-reverse-flick process. The last (flicking) step is functionally equivalent to a tumble in E. coli, allowing the bacterium to randomly select a direction, and a new cycle ensues. Despite the fact that run and reverse intervals, ∆ f and ∆ b , as well as the flicking angle ∆θ are stochastic, the 3-step Figure 1: Bacterial swimming trajectories of V. alginolyticus (a) and E. coli (b). The cells have been selected amongst many because they are more or less swimming in the focal plane, 50-100 µm above the glass coverslip. The starting points for both trajectories are indicated by the two large solid dots. The time lapse between adjacent dots are 0.067 s and 0.13 s for (a) and (b). The red and green segments in (a) designate the backward and forward swimming intervals, and transitions from backward to forward cause flicking, randomizing the swimming direction. Unlike a transition from forward to backward, which has a directional change ∆θ ≃ π (or backtracking), a backward to forward transition is random with ∆θ uniformly distributed between 0 and 180 o . (c) A hypothetical response function R(t) of E. coli based on Tu et al.'s model (6) is plotted as the dark curve, and the cell is stimulated at t = 0 with a brief pulse of attractant (red curve). Here, , where we set R 0 = 1, and used the typical E. coli methylation time (τ m = 3 s) and phosphorylation time (τ z = 0.5 s) (6,7).
cycle is deterministic and has been observed in different V. alginolyticus strains and in a swimming buffer with and without a chemical gradient (8). In a steady state without a chemical gradient, we found that the probability density functions P (∆ b ) and P (∆ f ) are statistically independent and have long exponential tails (or a Poissonian-like behavior) with the mean intervals τ b ≃ τ f ≃ 0.3 s. However, when a point source of chemoattractant is present, the cells can quickly migrate along the gradient and form a tight pack around the source.
The biochemical network that regulates the activity of E. coli motor is reasonably well understood (9). While this is not the case for V. alginolyticus, it cannot deter our progress because we know that even for very diverse microorganisms, such as E. coli and Bacillus subtilis that are roughly one billion years apart according to the recently constructed phylogenetic tree (10), the fundamental mechanism of regulation is still similar, i.e. a ligand binding to a receptor triggers a cascade of chemical reactions. The end product of the reaction is a chemically modified protein, called the response regulator (CheY-P), that binds to the motor, causing it either to rotate CCW (B. subtilis) or CW (E. coli ). The basic aim of different microorganisms is also the same, namely guided by chemical signals, the cell is directed towards the source of chemoattractant and away from chemorepellent. According to the phylogenetic tree (10), V. alginolyticus appears to be much closer to E. coli than B. subtilis, suggesting that there is much in common between these different bacterial species. Indeed in V. alginolyticus, one can identify chemotaxis genes that are largely homologous to E. coli with the exception of cheV that is absent in E. coli but is present in B. subtilis. A recent study moreover showed that the phosphorylated CheY in V. alginolyticus causes the polar flagellar motor to reverse the direction from CCW to CW, similar to E. coli (11).
It is clear that the 3-step swimming pattern is significantly different from the wellstudied 2-step swimming pattern of run and tumble, and it has strong implications for bacterial chemotaxis, which can be characterized by an effective diffusion coefficient D and a drift velocity V in the presence or absence of a chemical gradient. The calculation below illustrates that cells executing the 3-step swimming pattern can exhibit rich chemotactic behaviors, and the variations can be acted on by natural selection so that a particular response emerges. Below we will illustrate these new aspects of bacterial chemotaxis based on our findings of the 3-step process.
Similar to cells of E. coli, the flagellar motor of V. alginolyticus has two lifetimes for the state of rotations: one (τ f ) for the CCW interval and one (τ b ) for the CW interval, where the subscripts f and b stand for forward and backward swimming, respectively. To modulate their chemotactic behaviors, these lifetimes are affected by the local concentration of chemoeffectors and cells' adaptation mechanism. Unlike E. coli, however, CW rotation in V. alginolyticus causes the cell to backtrack. Both swimming intervals are expected to depend on the ligand concentration c(t), which we assume to be chemoattractant. For small c(t), we assume that a linear response is applicable and hence, where τ f and τ b are the steady-state values, and R f (t) and R b (t) are the memory (or the response) functions, which are not necessarily the same for the two swimming intervals.
In the above, an exposure to the ligand causes the forward lifetime to increase, and is consistent with our observations in V. alginolyticus (8). Linearity of Eqs. 1 and 2 suggests that it is possible to examine one delay time θ at a time and sum up all possible delays at the end. Following de Gennes, we write R s (t) = α s δ(t − θ), where the strength of the response α s (s = f, b) has the dimension of volume. Next, we consider a cell moving in a chemical gradient as depicted in Fig. 2. Our aim is to calculate the displacement x i along the gradient in one cycle, ∆ f +∆ b , which leads to a mean drift velocity V =x i /(τ f +τ b ) after averaging over ∆ f and ∆ b . Because a cell randomizes its swimming direction at the end of the backward interval by a flick, the motions in two consecutive cycles are uncorrelated. This allows us to place the origin of time (t = 0) at the beginning of the forward run.
Assuming that the forward run time is Poisson distributed, the surviving probability of a cell swimming forward up to ∆ f is given by, and the probability that it stops immediately after ∆ f is −∂P f (∆ f )/∂∆ f . Likewise, the surviving probability of a cell swimming backwards from ∆ f to ∆ f + ∆ b is given by, and the stopping probability at the end of the backward run is It follows that the net mean displacement in one cycle is given by, wherex f i andx bi represent respectively the mean displacement during the forward (∆ f ) and the backward (∆ b ) swimming interval, and ... designates the angular average for v f i and v bi . For the linear gradient depicted in Fig. 2, the concentration experienced by the cell can be represented as Since c 0 is determined by the velocity in the previous cycle, it does not contribute to the above integrations after angular averaging. Although the calculation of Eq. 5 is tedious, which is given in Appendix A, the final result is straightforward: The first term in the curly bracket of Eq. 6 is the displacement during the forward interval, and the second term is the displacement during the backward interval. It is noteworthy that during the second interval, there is a cross term proportional to , which results from the delay, i.e., even though the cell is moving backwards, in the early episode of that interval, the cell still remembers the concentration sensed during the previous forward swimming. This gives rise to anti-correlation, since v f i v bi < 0, that contributes to a negative displacement. This important correlated motion adds richness to bacterial chemotaxis and is what makes V. alginolyticus behave differently from E. coli. We noted that Eq. 6 yields the result It is interesting that even when there is no memory, the cross term survives because there is no direction randomization after a forward run. Moreover, the total displacement during the backward interval, can contribute positively or negatively to the displacement, depending on the mean lifetimes τ f and τ b , and the swimming speeds v f and v b . We found the swimming pattern of V. alginolyticus is approximately symmetric with where the subscript f stands for the forward run (or CCW rotation) and φ 0 = τ CCW /(τ CCW + τ CW ) is the CCW bias. Since near a steady state φ 0 ≃ 0.5 or ∼ 0.8 according to Refs. (3) and (12), respectively, E. coli cells produce a smaller displacement than V. alginolyticus within one swimming cycle if everything else is equal. Using v f i = −v bi = v i and summing up all possible delays, we found from Eq. 6 that the mean displacement is given bȳ The average drift speed in the gradient is V (≡ κ∇c) =x i /(τ f + τ b ), which allows the chemotaxis coefficient κ to be calculated. In E. coli, κ is proportional to the diffusion For an organism exhibiting the 3-step swimming pattern, the diffusivity is given by: This calculation leads to two possible scenarios (or fundamental hypotheses) for bacterial chemotaxis: (i) independent and (ii) shared chemosensing. In the first case, the response functions in the forward and backward intervals are uncorrelated, i.e., R f (θ) and R b (θ) have different functional forms, so that the sensing system breaks the time reversal symmetry. In order to achieve such a control, the flagellar motor cannot only passively receive signals from the chemotaxis network but instead the status of the motor must be made known to the chemotaxis regulatory network. This may be attained either by the flagellar motor being a part of the regulatory network or by a feedback signal via a protein that can reset the chemotactic response. In short, there will be a back flow of information from the motor to the chemotaxis network in addition to the normal chemotaxis regulation. To optimize the drifting velocity, we applied a variational principle to Eq. 9, which is delineated in Appendix B. We used the constraints that R f (θ) and It is evident from the optimization procedure that in order to attain the maximum possible drifting speed, the forward response function R f (θ) should be monophasic but the backward response function R b (θ) can be either monophasic or biphasic, depending on the ratio of the two lifetimes, β ≡ τ b /τ f . Fig. 3 displays for different values of β = 0.8, 1.2, 1.5, 1.8, and 2.4. The figure shows that the biphasic character of R b (θ) becomes more pronounced as β decreases towards unity but dissappears altogether for β < 1, where the response is negative for all θ. An analysis shows that the biphasic response occurs in a narrow range of β (1 ≤ β ≤ 2), and outside this range the response is always monophasic. This behavior is understandable since when τ b is shorter than τ f , the backward interval is strongly influenced by the signal sensed in the previous forward interval due to the memory effect. To deal with this inconsistency between sensing and motility, the optimal strategy is a negative monophasic response as depicted by the purple curve (β = 0.8) in Fig. 3. On the other hand, when τ b is longer than τ f , the cell would have consistent sensing and motility so that a monophasic positive response is more favorable, which is shown by the red curve (β = 2.4) in Fig. 3. In the limiting case τ b ≫ τ f or τ b ≪ τ f , Eqs. 9, 10, and 11 make it clear that the chemotactic coefficient κ is dominated respectively by the backward or the forward swimming interval. The situation is formally equivalent to E. coli chemotaxis, where the monophasic response is optimal for a fast migration in a linear chemical gradient as was concluded by de Gennes (5).
In the case of shared chemosensing, the bacterium uses a single response function R(θ), albeit the amplitudes of the responses may be different in the two directions, R f (θ) = R b (θ)/γ = R(θ). A simple reason for γ = 1 could be due to different swimming speeds v f and v b , but other possibilities may also exist. For this type of sensing, there is no breaking of time reversal symmetry since the chemotaxis network processes information received during the forward and the backward interval equally, and there is no need for a back flow of information. Using Eq. 9, we found: Applying the variational principle again (see Appendix B), we found that the drift velocity is optimized by the following response function As displayed in Fig. 4, R(θ) can be monophasic or biphasic depending on γ as well as the time ratio β ≡ τ b /τ f . The biphasic regime is bounded by 1−β β ≤ γ < 1 β(1−β) for 0 ≤ β < 1 and γ ≥ 0 for 1 < β ≤ 2, which is displayed in the inset of Fig. 3. The inset shows that the parameter space (γ, β) consists of four different regimes with I, III, and IV being monophasic and II biphasic. Our theory hence predicts that if a bacterium uses a single response function, for very short (β ≪ 1) or very long (β ≫ 1) backward swimming intervals, the biphasic response is not a good chemotactic strategy for migration in a linear chemical gradient. The biphasic response emerges only when τ f and τ b being close (or β ≃ 1), which is the case in V. alginolyticus (8). It is conspicuous that in the limits β → 1 and γ → 1, R(θ) calculated using Eq. 13 is identical to the solution of a critically damped harmonic oscillator, which has the interesting property of ∞ 0 (1 − θ/τ ) exp(−θ/τ )dθ = 0, i.e., the response is "precisely" adaptive.
The above two hypotheses are testable by laboratory experiments where the bacteria are subject to a defined chemical stimulation, and one measures the switching rate s(t) = 2/(τ f (t) + τ b (t)) and the forward swimming bias φ(t) = τ f (t)/(τ f (t) + τ b (t)) as a function of time. For a weak stimulation, the above calculation allows us to find, where s 0 ≡ 2/(τ f + τ b ) and φ 0 ≡ τ f /(τ f + τ b ) are the steady-state switching rate and the forward bias, respectively. The expressions are significantly simplified if the perturbation is δ-in-time, c(t) = c ′ δ(t), and they are given by, The calculation shows that if the second scenario is true and R f (t) ≃ R b (t), the forward bias will be weakly dependent on the time t, and the switching rate is simply given by . However, if the first scenario is true, the measured s(t) and φ(t) can be used to find the response function R f (t) and R b (t) using Eqs. 16 and 17. In this case, the following simple relations result, An alternative and perhaps more direct way to find R f (t) and R b (t) is to perform conditional stimulation for individual cells. The bacterium can be either tethered to a surface, such as in Block et al.'s experiment (3), or freely swimming, as in Khan et al.'s experiment (13). For tethered cells, one can apply a pulse of chemoattractant at the moment the motor switches from CW (CCW) to CCW (CW), and record the subsequent swimming interval ∆ 1f (∆ 1b ), where the subscript 1 emphasizes the interval before the first switch. By counting the switching events up to time t, one can construct a cumulative PDF (normalized by the total number of cells) Ψ s (t), and the time-dependent switching rate can be obtained according to τ −1 , where s = f or b. For freely swimming cells, one can use photo-active serine, which is an attractant to V. alginolyticus, to stimulate cells. If the first scenario is true, one should find that τ −1 f (t) and τ −1 b (t) have different time dependence or equivalently R f (t) and R b (t) have different functional forms. However, if the second scenario is true, there should be not much difference between τ −1 f (t) and To conclude, the 3-step motility pattern of V. alginolyticus discovered in our recent experiment (8) allows significant variations in bacterial chemotactic behavior. These variations can be acted on by natural selection and give rise to distinct phenotypes observed in the wild. Compared to the 2-step swimming pattern of E. coli, cells of V. alginolyticus can engage in chemosensing and migration in both the forward and the backward swimming intervals, and hence their "duty cycle" is ∼ 100% as compared to ∼ 50 − 80% in E. coli (3,12). An important aspect in 3-step chemotaxis is backtracking that gives those bacteria heading down a gradient an opportunity to re-exploit what they find a moment earlier. In our opinion, the full duty cycle, backtracking, and flicking are defining characteristics of V. alginolyticus. These significant niches are likely selected for by the ocean environment where a quick response to transitory signals is important. We showed that for a swimmer executing the cyclic 3-step motility pattern, a biphasic response arises naturally without the need to invoke cell localization as suggested for E. coli (1). Moreover, we showed that the biphasic response is most effective when the forward τ f and the backward τ b swimming intervals are comparable. This makes biological sense since a brief forward or a brief backward interval contributes little to motility, and consequently a monophasic response is sufficient for migration. This also raises the interesting question why the non-motile CW interval in E. coli is so long, taking up at least 20% of the duty cycle. If tumbling is just to change the direction, would not it be better if CW interval is shorter? An interesting possibility is that the ancestral cell that gave "birth" to E. coli and V. alginolyticus was a 3-step swimmer. However, when E. coli became specilized in a different environment, which favored multiple flagella for motility, they gave up backtracking and flicking, resulting in a tumbly movement. In this view, then, it is not surprising that E. coli 's tumbling interval is long and its chemotactic response is biphasic.
Based on motility alone, we propose two different mechanisms, independent and shared chemosensing, by which cells of V. alginolyticus can optimize their migration speed in a linear gradient. Interestingly, the biphasic response appears in both types of chemotactic strategies. We know very little at present how the chemotaxis network of V. alginolyticus regulates the 3-step motility pattern. However, based on our calculation, it is likely that a biphasic response is also adopted by V. alginolyticus, and it is awaiting to be verified in future experiments. Finally, it would be interesting to generalize the above calculation to situations where the chemical landscape is constantly changing, such as chemical waves (14).
This work is supported by the NSF under the grant no. dmr-bp0646573.
The first Heaviside function confines the integral to t < 0, and since v ′ bi v f i = 0, there is no contribution from this term. The integration constrained by the second Heaviside function yields,x This equation is identical to that found by de Gennes when he calculated the drift velocity for E. coli cells (5). The second part of Eq. 5 is more complicated because one has to take into account more possibilities. Again, we used integration by parts to obtain, Let the first term in the above equation bex bb bi and the second term bex bf bi . Since P f (0) = 1 and P b (∆ b , 0) = P f (∆ b ), it follows that the integration in the first term is identical to Eq. 20 with the replacement of the subscript f by b. This yields, Now, lets examine the anti-correlation termx bf bi , which corresponds to the situation when the bacterium swims down the gradient but it still keeps its "old good memory". Dropping the nonlinear terms in concentration c, we found, When integrating over ∆ f , there are two possibilities for the lower limit of the t-integration, i.e. either ∆ f − θ < 0 or ∆ f − θ ≥ 0. These will be delimited by the Heaviside functions as before. For each of these cases, while integrating over ∆ b , there are additional possibilities for the upper limit of the t-integration. For the first case, when ∆ f − θ < 0, there are three possibilities: corresponding to the regimes I ′ , I, and II in Fig. 2, respectively. However, since motion is uncorrelated after a flick or v ′ bi v bi = 0, the first possibility does not contribute to the displacement. In the second case, when ∆ f − θ ≥ 0, there are two additional possibilities: (iv) corresponding to the regimes I and II in Fig. 2, respectively. The corresponding time integrals for the above four possibilities (ii-v) are given by where c 1 ≡ c 0 + ∇c v f i ∆ f . Using the above expressions, we take the derivative with respect to ∆ f to obtain, (v) Again, using the Heaviside function to represent these four non-trivial possibilities, we have the following identity Substituting this equation into Eq. 25, we found, where (ii), (iii), (iv), and (v) are the terms given in Eq. 27. The four integrations in the above equation are delimited by different combinations of Heaviside functions, yielding different lower and upper integration limits for each integral. Designating these integrals