A connection between bacterial chemotactic network and optimal filtering
aa r X i v : . [ q - b i o . CB ] M a y A connection between bacterial chemotactic network and optimal filtering
Kento Nakamura and Tetsuya J. Kobayashi ∗ Department of Mathematical Informatics, Graduate School of Information Science and Technology, the University of Tokyo (Dated: May 28, 2020)The chemotactic network of
Escherichia coli has been studied extensively both biophysically andinformation-theoretically. Nevertheless, the connection between these two aspects is still elusive.In this work, we report such a connection by showing that a standard biochemical model of thechemotactic network is mathematically equivalent to an information-theoretically optimal filteringdynamics. Moreover, we demonstrate that an experimentally observed nonlinear response relationcan be reproduced from the optimal dynamics. These results suggest that the biochemical networkof
E. coli chemotaxis is designed to optimally extract gradient information in a noisy condition.
Living things have developed sensory systems to be-have appropriately in changing environments. One of themost-analyzed such systems is the sensory system of
Es-cherichia coli for chemotaxis. In
E. coli chemotaxis, a cellobtains information of a spatial gradient of a ligand fromthe temporal change in the ligand concentration that itexperiences by swimming in the gradient. An
E. coli cellcan sense a positive change in the ligand concentrationwhen it swims along the direction of the gradient andvice versa. The swimming trajectory of
E. coli consistsof a series of ballistic swimming called run interruptedwith random reorientations of direction called tumbling.By inhibiting the frequency of tumbling when it sensesa positive change in an attractant concentration, the
E.coli cell can elongate the run length toward the directionof the higher concentration.The mechanism of the sensory system has been inten-sively studied both experimentally and theoretically. Ex-perimental studies have revealed the response of
E. coli tovarious temporal profiles of concentration by measuringbehaviors of motor rotation [1, 2] and signaling molecules[3, 4]. Theoretical studies have proposed and analysedbiochemical models that can reproduce properties of theexperimentally observed responses such as high sensitiv-ity to weak changes in concentration [4–7] and sensoryadaptation [8]. Based on these works, Tu et al proposeda simplified biochemical model [9], which can explain var-ious aspects of the responses simultaneously [10]. Thisbiochemical model has been widely employed for variouspurposes such as analysis of sensory-motor coordination[11], fold-change detection [12, 13], and thermodynamicsof sensory adaptation [14].In Tu’s model [9], the sensory system consists of re-ceptor complexes, each of which takes either active orinactive state. Active receptors send a signal via media-tor proteins and control the rotation of flagellar motors.The ratio of active receptors, termed receptor activity a t , is subjected to a feedback regulation through recep-tor modification characterised by methylation level m t .The receptor activity a t is determined by the free energy ∗ Also at Institute of Industrial Science, the University of Tokyo;[email protected] difference f t between active and inactive states: a t = 11 + exp( f t ) . (1)The free energy difference f t comprises the additive effectof the methylation level m t and of the ligand concentra-tion [ L ] t as f t = N ( − αm t + log[ L ] t ) , (2)where we omit a constant term and N, α > N specifies the receptor cooper-ativity producing high sensitivity [4, 6, 7]. The methy-lation level m t is modulated by the receptor activity a t as d m t d t = F ( a t ) , (3)where F is assumed to be a monotonically decreasingfunction. Since d a t / d m t > F ′ ( a t ) <
0, the dy-namics of the methylation level m t with the function F constitutes a negative feedback regulation over the re-ceptor activity a t . Due to the negative feedback, thisbiochemical network displays the sensory adaptation [8],that is, when the concentration [ L ] t is stationary, the re-ceptor activity converges to a single value ¯ a such that F (¯ a ) = 0 which is independent of background concentra-tion.Although the biochemical model captures the integralparts of the sensory system and its behaviors, there isroom for discussion from the view point of noise toler-ance. Because the sensory system relies on stochasticligand-receptor interactions and receptor modifications,sensing signal inevitably contains noise. This noise wouldcause a fatal influence on the chemotactic performancebecause it can bury the actual temporal changes in con-centration and could end up with misdirections of themotor control. Therefore, the sensory system of E. coli is expected to have a certain noise filtering property, andseveral works have investigated impacts of noise in in-formation transmission and favorable traits for noise fil-tering [16]. However, these works focused on linear re-sponse by ignoring the underlying biochemical networkand resultant nonlinear properties of the
E. coli sensorysystem. Even though some others considered a possi-ble biochemical implementation of an ideal noise-immunesystem based on nonlinear filtering theory [17], the corre-spondence with actual biological systems, especially thatof the gradient sensing in chemotaxis, is still elusive.In this paper, we utilize nonlinear filtering theory to de-rive noise tolerant gradient sensing dynamics and demon-strate its biochemical implementation in
E. coli ’s cell. Inparticular, we find that the derived ideal noise-filteringsystem excellently coincides with Tu’s biochemical modelfor the
E. coli sensory system [9] and reproduces a non-linear response relation measured experimentally.As a minimal model of the temporal gradient sens-ing, we consider a run-tumble motion of
E. coli on onedimensional axis along with monotonically increasing lig-and concentration. This assumption is mainly due to thelimited capacity of the cell that may not be able to rec-ognize the three dimensional physical space. Let ξ t ∈ R and X t ∈ {− , +1 } be the location and the direction ofswimming at time t ∈ [0 , ∞ ). We assume that an E.coli cell runs ballistically with a constant speed v > ξ t / d t = vX t and that each run and its directionis interrupted by a stochastic tumbling motion. By ap-proximating the tumbling motion by an instantaneousevent[18], we model the random changes in direction X t due to tumbling with a continuous-time Markov chain:d p t d t = (cid:18) − r − r + r − − r + (cid:19) p t , (4)where p t = ( P ( X t = +1) , P ( X t = − T , and r + and r − are the time-independent transition rates from − −
1, respectively. Note that thetransition rate of direction X t would be smaller than therate of tumbling event because each tumbling does notalways lead to the flipping of the direction.Next, we assume that the ligand concentration dependsexponentially on the location as [ L ] t ∝ exp( cξ t ) where c > Y t = − log[ L ] t − √ σW t (5)where W t is the standard Wiener process and σ is theintensity of noise. It should be noted that W t can alsobe interpreted as the noise from methylation [19] becausethe methylation level m t additively appears in Eq. (2).By applying the nonlinear filtering theory under theabove settings and assumptions [20], we can derive thefollowing stochastic differential equation asd Z t d t = − R ( Z t − ¯ p ) + KZ t (1 − Z t ) ◦ d Y t d t , (6)where ◦ is the Stratonovich integral (See supplementarymaterial (SM) for details of derivation). This equation describes the posterior probability Z t = P ( X t = − | Y t ) of the descending direction given the time series ofthe noisy sensing Y t := { Y t ′ | t ′ ∈ [0 , t ] } when its pa-rameter values matches those of tumbling, run, gradi-ent, and noise as R = R OPT := r + + r − , ¯ p = ¯ p OPT := r − / ( r + + r − ), K = K OPT := 2 vc/σ .Under this set of the optimal parameter values, the firstterm represents the prediction based on a prior knowl-edge about switching dynamics of direction X t (Eq. (4)).Thereby, without the second term (sensing signal), Z t converges to the stationary probability of the direction¯ p for t → ∞ . The second term corresponds to the up-date of the posterior by new observation (Eq. (5)). Theoptimal gain of this term, K OPT , describes the signal-to-noise ratio because σ and 2 vc specifies the noise intensityand the steepness of the temporal change in the ligandconcentration during swimming, respectively. We callthe dynamics of Z t described by Eq. (6) the filteringdynamics hereafter.Next, we reveal the relation between the filtering dy-namics and the biochemical network of E. coli chemotaxisby demonstrating that Eq. (6) can be equivalent to Eqs.(1),(2), and (3) if noise is neglected.To this end, we introduce a coordinate transform fromthe posterior probability Z t to the log-likelihood ratio θ t := log Z t / (1 − Z t ). From the chain rule for deriva-tives, d θ t / d t = (d θ t / d Z t )(d Z t / d t ), we obtain the follow-ing equivalent representation of the filtering dynamics:d θ t d t = R Z t − ¯ pZ t (1 − Z t ) − K ◦ d Y t d t . (7)By further defining a new variable µ t for the predictiondynamics as d µ t d t := − Rκ Z t − ¯ pZ t (1 − Z t ) , (8)then we can formally integrate Eq. (7) as θ t = − κµ t + K (cid:2) log[ L ] t + √ σW t (cid:3) . (9)where we use Eq. (5) and κ > Z t in Eq. (8) can be obtained by the inversetransformation from θ t to Z t : Z t = 11 + exp( θ t ) . (10)These transformations unveil that Eqs. (10),(9), and(8) for the filtering dynamics are equivalent to Eqs.(1),(2), and (3) for the biochemical model of E. coli chemotaxis, respectively (see also table S1 in SM for com-parison).The posterior probability Z t corresponds to the re-ceptor activity a t and they are both described by thesigmoidal function of θ t and f t , respectively. The log-likelihood ratio θ t is determined by the logarithm of theligand concentration [ L ] t and the prediction term µ t ,which corresponds to the dependence of the free energydifference f t on the ligand concentration [ L ] t and themethylation level m t in Eq. (2). Finally, the dynamicsof prediction term µ t corresponds to that of the methy-lation level m t .Because the right-hand-side of Eq. (8) is a decreasingfunction of Z t in the same way as the feedback func-tion F ( a t ) of m t , µ t works as a negative feedback com-ponent to Z t . Even though F ( a t ) in Tu’s model can-not be determined biochemically but inferred only ex-perimentally, the filtering dynamics provide a concretefunctional form for the feedback function, F OPT ( Z ) := − ( R/κ ) · ( Z − ¯ p ) / ( Z (1 − Z )). Thus, if E. coli has developedthe sensory system being tolerant to sensing noise nearoptimally, the feedback function F describing the methy-lation dynamics can have a similar form as F OPT . Totest this expectation, we compare the feedback function F EXP inferred experimentally by a FRET measurement[21] with the theoretically predicted F OPT by adjustingtwo free parameters
R/κ and ¯ p . Figure 1 shows a notableagreement between the experimental data and theoreticalprediction. Both F EXP and F OPT share a characteristicnonlinearity; a gentle slope around a = 0 . a = 1. This result implies that the E. coli chemotactic network is designed structurally to be robustto the sensory noise. In addition, because ¯ p in F OPT cor-responds to the stationary probability that the directionof swimming is down the gradient, the parameter values¯ p ≈ .
28 obtained by fitting implies that
E. coli has aprior expectation that it likely swims up the gradient.We further investigate whether the biochemical pa-rameters observed experimentally in laboratory environ-ments can satisfy the optimality in terms of filtering.From the fitting of F OPT to F EXP , we have
R/κ ≈ . × − . κ can be estimated as κ = αN ≈
12 bycomparing Eq. (2) and Eq. (9) and by employing a pre-vious estimate of α and N [21]. Thus, R is calculated as R ≈ . × − . In contrast, the optimal R OPT can beestimated from R OPT = r + + r − and measurements oftumbling rate as 10 − . ≤ R OPT ≤ s − [2, 22]. Thus,the obtained biochemical parameter R is much smallerthan the estimate R OPT from tumbling measurements.This discrepancy may be attributed to three possi-bilities: First, experimental conditions for the measure-ments of tumbling rate might not capture a wild condi-tion where
E. coli cells are supposed to perform chemo-taxis. Recent studies suggest that swimming behavior inpolymeric solutions or soft agar is different from that un-der a liquid condition used in most experiments [23]. Inparticular, the tumbling frequency is shown to decreasewith addition of polymeric molecules due to remodelingof signaling pathway downstream of sensory system orpossibly due to motor load. In such a case, R OPT maytake smaller value. Second, the values of R might be un-derestimated because of the difficulty in estimating thebiochemical parameter N . Although we used an estimate N = 6 in previous studies [7, 9, 21], other estimates of N are larger, N = 15 ∼
20 [7, 24]. The last possibilityis that the system is not or cannot be always optimized F ( a ) filteringexperiment FIG. 1. Theoretically derived F OPT (red curve) fitted to theexperimentally obtained F EXP (black points) [21]. F OPT inthe figure is obtained by modulating two parameters, R and¯ p , as R/κ ≈ . × − and ¯ p ≈ .
28 (see also SM for thefitting procedure). at the level of parameter values, though it is so at thelevel of network structure. By considering the correspon-dence of N with the gain K OPT , which is determined bythe speed of swimming, steepness of the gradient, andintensity of sensing noise, N should not be fixed at cer-tain value but be variable depending on environmentalsituations. Several studies suggested that N as well asother parameters are diversified in a population of cellsfor hedging environmental uncertainties [25].To perform chemotaxis under the limitation in pa-rameter adjustment, the robustness against the mis-match of parameters could be beneficial. We inves-tigate whether such robustness is endowed by exam-ining the filtering dynamics with misspecified param-eter values of K . We measure the performance ofthe dynamics using mean square error (MSE) definedas h T R Tt =0 { X t − (1 − Z t ) } d t i / in which 1 − Z t =1 − P ( X t < | Y t ) = E [ X t | Y t ] holds for the opti-mal parameter set. We define a reference value of K as K ref := N = 6 according to the correspondence between K and N . We set swimming speed to a physiologicallyrelevant value: v = 20 µ m · s − . The rates of directionalchanges are determined as r + = R (1 − ¯ p ) , r − = R ¯ p so thatthe values of R and ¯ p obtained by fitting in Fig. 1 be-come optimal. We define the reference of the steepness ofgradient as c ref := 10 − µ m − by taking into account con-ditions in previous simulation studies [11]. We also de-fine the reference of noise intensity as σ ref := 2 c ref v/K ref such that the reference parameter K ref is optimal un-der c = c ref and σ = σ ref . Note that K ref is also opti-mal on the half-line, ( σ, c ) = η ( σ ref , c ref ) , η >
0, because2 vc/σ = 2 vc ref /σ ref = K ref holds on it.Figure 2 shows MSEs of Eq. (6) for different K asfunctions of σ with fixed c = c ref (A) and as functionsof c with fixed σ = σ ref (B). The error with fixed K is (A) (B)(C) (D) FIG. 2. MSE of the filtering dynamics as a function of σ with fixed c = c ref (A), as a function of c with fixed σ = σ ref (B), as a function of K and c with fixed σ = σ ref (C), andas a function of σ and c with fixed K = K ref (D). Curves in(A) and (B) represent MSEs with fixed parameter K = K ref (blue) and with the optimal parameter K = K OPT = 2 vc/σ (red). White lines in (C) and (D) represent the parameterregion on which the parameter K is set optimal i.e. 2 vc/σ ref = K (C) and 2 vc/σ = K ref (D). always greater than or equal to that with K adjustedto K OPT . For each fixed gain K , MSE monotonouslyincreases as the signal-to-noise ratio (SNR) decreases ei-ther by the increase in the noise intensity σ (Fig. 2(A))or by the decrease in gradient steepness c (Fig. 2(B)),indicating that greater SNR than optimal one never im- pair the performance of the dynamics for any K . We cansee a similar trend in Fig. 2 (C) and (D). These resultsindicate that even under the misspecification of K asso-ciated with parameters σ and c , the filtering dynamicsstill reliably and robustly estimate temporal gradient ifthe change in σ and c is one that increases SNR.Small value of gain K is optimized to a low SNR situ-ation, and variation of MSE between low and high SNRsis small (Fig. 2). In contrast, large K adjusted to ahigh SNR one shows a significant variation in MSE be-tween low and high SNR cases. This means that low K can work moderately well for most of conditions whereaslarge K can work much better if the environmental SNRis large enough at the cost of lower performance underlow SNR situations. Thus, K modulates the balance ofrisk-averting and -taking strategies of sensing.The growth-dependent variability of K can coordi-nate such risks at the level of population [26]. More-over, N , which biochemically corresponds to K , is sug-gested to vary temporally at the single-cell-level [24, 27]via a receptor cluster rearrangement. The integration ofbiochemical modeling and optimal filtering theory couldwork for further analysis of such a gain adaptation ofcells. This approach may also apply to other sensorysystems with allosteric receptors and a negative feedback,e.g., G protein-coupled receptors for vision and EGF re-ceptor in animal cells, whose models can be reduced tosimilar biochemical models to Tu’s [9]. 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