A PATHWAY-BASED MEAN-FIELD MODEL FOR E. COLI CHEMOTAXIS: MATHEMATICAL DERIVATION AND ITS HYPERBOLIC AND PARABOLIC LIMITS

Transcription

1 A PATHWAY-BASED MEAN-FIELD MODEL FOR E. COLI CHEMOTAXIS: MATHEMATICAL DERIVATION AND ITS HYPERBOLIC AND PARABOLIC LIMITS GUANGWEI SI, MIN TANG, AND XU YANG Abstract. A pathway-based mean-field theory PBMFT that incorporated the most recent quantitatively measured signaling pathway was recently proposed for the E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 19 1, 4811]. In this paper, we formally derive a new kinetic system of PBMFT under the assumption that the methylation level is locally concentrated, whose turning operator takes into account the dynamical intracellular pathway, and hence is more physically relevant. We recover the PBMFT proposed by Si et al. as the hyperbolic limit and connect to the Keller-Segel equation as the parabolic limit of this new model. We also present the numerical evidence to show the quantitative agreement of the kinetic system with the individual based E. coli chemotaxis simulator. 1. introduction The locomotion of Escherichia coli E. coli presents a tumble-and-run pattern [5], which can be viewed as a biased random walk process. In the presence of chemoeffector with a nonzero gradient, the suppression of direction change tumble leads to chemotaxis towards the high concentration of chemoattractant [1, 4]. A huge amount of efforts has been made to understand the chemotactic sensory system of E. coli e.g. [11, 18, 3, 34]. The chemotactic signaling pathway belongs to the class of two-component sensory systems, which consists of sensors and response regulators. The chemotactic sensor complex is composed of transmembrane chemo-receptors, the adaptor protein CheW, and the histidine kinase CheA. The response regulator CheY controls the tumbling frequency of the flagellar motor [19]. Adaptation is carried out by the two enzymes, CheR and CheB, which control the kinase activity by modulating the methylation level of receptors [34]. Because of the slow adaptation process, the receptor methylation level serves as the memory Date: April 9, 14. G.S. was partially supported by NSF of China under Grants No and No and the MOST of China under Grants No. 9CB9185 and No. 1AAA7. M.T. was partially supported by Natural Science Foundation of Shanghai under Grant No. 1ZR14454 and Shanghai Pujiang Program 13PJ147. X.Y. was partially supported by the Regents Junior Faculty Fellowship of University of California, Santa Barbara. G.S. would like to thank Yuhai Tu for valuable discussions and Tailin Wu for his early work on simulation. 1

2 GUANGWEI SI, MIN TANG, AND XU YANG of cells in a way that the cells effectively run or tumble by comparing the receptor methylation level to local environments. In the modeling literature, bacterial chemotaxis has been described by the Keller- Segel K-S model at the population level [3], where the drift velocity is given by the empirical functions of the chemoeffector gradient. It has successfully explained chemotactic phenomena in slowly changing environments [31], however failed to predict them in rapidly changing environments [36], including the so-called volcano effects [1, 8]. Besides that, the K-S model has also been mathematically proved to present nonphysical blowups in high dimensions when initial mass goes beyond the critical level [6 8]. In order to understand bacterial behavior at the individual level, kinetic models have been developed by considering the velocity-jump process [3,1,3], and the K-S model can be derived by taking the hydrodynamic limit of kinetic models e.g. [1, 17]. All the above mentioned models are phenomenological and do not take into account the internal signal transduction and adaptation process. It is especially hard to justify the physically relevant turning operator in the kinetic model. Nowadays, modern experimental technologies have been able to quantitatively measure the dynamics of signaling pathways of E. coli [,13,6,9], which has led to the successful modeling of the internal pathway dynamics [4, 5, 33]. These works made possible the development of predictive agent-based models that include the intracellular signaling pathway dynamics. It is of great biological interest to understand the molecular origins of chemotactic behavior of E. coli by deriving population-level model based on the underlying signaling pathway dynamics. In the pioneering work of [15, 16, 35], the authors derived macroscopic models by studying the kinetic chemotaxis models incorporating linear models for signaling pathways. In [7], the authors developed a pathway-based mean field theory PBMFT that incorporated the most recent quantitatively measured signaling pathway, and explained a counter-intuitive experimental observation which showed that in a spatialtemporal fast-varying environment, there exists a phase shift between the dynamics of ligand concentration and center of mass of the cells [36]. Especially, when the oscillating frequency of ligand concentration is comparable to the adaptation rate of E. coli, the phase shift becomes significant. Apparently this is a phenomenon that cannot be explained by the K-S model. In this paper, we study the PBMFT for E. coli chemotaxis based on kinetic theory. Specifically we derive a new kinetic system whose turning operator takes into account the dynamic intracellular pathway. The difference of this new system is that, compared with those kinetic models in [3, 1, 3], neither the turning operator nor the methylation level depend on the chemical gradient explicitly, which is more consistent with the recent computational studies in [7]. Besides, all parameters can be measured by experiment and quantitative matching with experiments can be

3 A MEAN-FIELD MODEL FOR CHEMOTAXIS 3 done. The key observation here is that, the methylation level is locally concentrated in the experimental environment. We formally obtain the Keller-Segel limit in the parabolic scaling and the PBMFT proposed in [7] in the hyperbolic scaling of the kinetic system, by taking into account the disparity between the time scales of tumbling, adaptation and experimental observation. The assumption on the methylation difference and the quasi-static approximation on the density flux in [7] can be understood explicitly in this new system. We also verify the agreement of the kinetic system with the signaling pathway-based E. coli chemotaxis agentbased simulator SPECS [] by the numerical simulation in the environment of spacial-temporal varying ligand concentration. The rest of the paper is organized as follows. We introduce the pathway-based kinetic model incorporating the intracellular adaptation dynamics in Section. In Section 3, assuming the methylation level is locally concentrated, we are able to derive the kinetic system independent of the methylation level in one dimension. Furthermore, the modeling assumption will be justified both analytically and numerically. By Hilbert expansion, Section 4. provides the recovery of the PBMFT model proposed in [7] in the hyperbolic scaling of the new system, illustrates why K-S model is valid in the slow varying environments, and show the numerical evidence of the quantitative agreement of the system with SPECS. The two-dimensional moment system is derived in Section 5, and we make conclusive remarks in Section 6.. Description of the kinetic model We shall start from the same kinetic model used in [7], which incorporates the most recent progresses on modeling of the chemo-sensory system [6, 33]. The model is a one-dimensional two-flux model given by.1. P + P = v P + = v P fap + fap za P + P, + za P + P. In this model, each single cell of E. coli moves either in the + or direction with a constant velocity v. P ± t, x, m is the probability density function for the cells moving in the ± direction, at time t, position x and methylation level m. The global existence results for the linear internal dynamic case has been established in [14] in one dimension as well as in [9] for higher dimensions. The intracellular adaptation dynamics is described by.3 dm dt = fa = k R1 a/a,

4 4 GUANGWEI SI, MIN TANG, AND XU YANG where the receptor activity am, [L] depends on the intracellular methylation level m as well as the extracellular chemoattractant concentration [L], which is given by.4 a = 1 + expne 1. According to the two-state model in [4, 5], the free energy is 1 + [L]/KI.5 E = αm m + f [L], with f [L] = ln. 1 + [L]/K A In.3, k R is the methylation rate, a is the receptor preferred activity that satisfies fa =, f a <. N, m, K I, K A represent the number of tightly coupled receptors, basic methylation level, and dissociation constant for inactive receptors and active receptors respectively. We take the tumbling rate function zm, [L] in [7],.6 z = z + τ 1 a/a H, where z, H, τ represent the rotational diffusion, the Hill coefficient of flagellar motor s response curve and the average run time respectively. We refer the readers to [7] and the references therein for the detailed physical meanings of these parameters. More generally, the kinetic model incorporating chemo-sensory system is given as below,.7 t P = v x P m fap + QP, z, where P t, x, v, m is the probability density function of bacteria at time t, position x, moving at velocity v and methylation level m. The tumbling term QP, z is.8 QP, z = zm, [L], v, v P t, x, v, m dv zm, [L], v, v dv P t, x, v, m, Ω where Ω represents the velocity space and the integral = 1, where Ω = Ω Ω denotes the average over Ω. zm, [L], v, v is the tumbling frequency from v to v, which is also related to the activity a as in.6. The first term on right-hand side of.8 is a gain term, and the second is a loss term. Ω Ω dv, 3. One-dimensional mean-field model In this section, we derive the new kinetic system from.1-. based on the the assumption that the methylation level is locally concentrated. This assumption will be justified by the numerical simulations using SPECS and the formal analysis in the limit of k R. To simplify notations, we denote + by in the rest of this paper.

5 A MEAN-FIELD MODEL FOR CHEMOTAXIS Derivation of the kinetic system. Firstly, we define the macroscopic quantities, density, density flux, momentum on m and momentum flux as follows, ρ + x, t = q + x, t = P + dm, ρ x, t = P dm; mp + dm, q x, t = mp dm; 3.3 J ρ = v ρ + ρ, J ρ = v q + q. The average methylation level of the forward and backward cells M + t, x, M t, x are defined as 3.4 M + = q+ ρ +, M = q ρ. For simplicity, we also introduce the following notations 3.5 Z ± = z M ± t, x, F ± = f a M ± t, x, [L] Assumption A. We need the following condition to close the moment system, m/m ± 1 P ± dm m/m ± 1 P ± dm P ± 1, dm m/m ± 1 P ± dm 1. Remark. Physically this assumption means, distribution functions P ± are localized in m, and the variation of averaged methylation is small in both moving directions ±. Integrating.1 and. with respect to m respectively yield the equation for ρ + and ρ such that ρ + ρ + 1 zap + dm zap dm ρ + v 1 zm + + z m M + P + dm M + zm + z m M P dm M ρ + 1 Z + ρ + Z ρ, ρ ρ = v + 1 zap + dm zap dm ρ v + 1 zm + + z M M +m + P + dm zm + z M M m P dm ρ = v + 1 Z + ρ + Z ρ.

6 6 GUANGWEI SI, MIN TANG, AND XU YANG where we have used Assumption A in the second step and the notations in 3.4, 3.5 in the third step. Similarly, multiplying.1 and. by m and integrating them with respect to m give the equation for q + and q respectively: q + q q + m fap + dm 1 mzap + dm q + v + fa m=m + + f m M + P + dm m=m + 1 M + Z + + mza M + m M + P + dm M Z + mza M m M P dm q + + F + ρ + 1 M + Z + ρ + M Z ρ, q = v m fap dm + 1 mzap + dm q v + fa m=m + f m M P dm m=m + 1 M + Z + + mza M + m M + P + dm M Z + mza M m M P dm M + Z + ρ + M Z ρ, mzap dm mzap dm q = v + F ρ + 1 where we have used an integration by parts and the definition of M + and M in 3.4 in the second step. Altogether, we obtain a system for ρ +, ρ, q + and q Z + ρ + Z ρ, ρ + ρ + 1 ρ ρ = v + 1 q + q + + F + ρ + 1 q q = v + F ρ + 1 Z + ρ + Z ρ, Z + q + Z q, Z + q + Z q. Remark. The Taylor expansion in m gives a systematical way of constructing high order systems. For example, we can introduce two additional variables e + x, t = m M + P + dm and e x, t = m M P dm, then construct a six equation system by approximating fm fm m=m ± + f m M ± + 1 f m=m m=m ± m M ±, ±

7 A MEAN-FIELD MODEL FOR CHEMOTAXIS 7 zm zm m=m ± + z m M ± + 1 z m=m m=m ± m M ±. ± 3.. Numerical Justification of Assumption A by SPECS. To justify the Assumption A, we simulate the distribution of m using SPECS in an exponential gradient ligand environment [L] = [L] expgx. SPECS is a well developed agent-based E. coli simulator that incorporates the physically measured signaling pathways and parameters []. In the simulation we introduced a quasi-periodic boundary condition: cells exiting at one side of the boundary will enter from the other side, and the methylation level is reset randomly following the local distribution of m at the boundaries. Using an exponential gradient ligand environment and this kind of boundary condition will lead to a well-defined distribution of cells methylation level. The steady state distributions are shown in Figure 1. In each of the subfigures, the horizontal and vertical axes represent the position and the methylation level respectively. As shown in Figure 1, the distribution of cells methylation level is localized, and becomes wider when G increases. M ± = mp ± dm are the average methylation levels for the right and left moving cells. One can also observe that M + < M in the exponential increasing ligand concentration environment. This can be understood intuitively by noticing that the up gradient cells with lower methylation level come from left while the down gradient cells with higher methylation level come from right. As shown in Fig. 1, in an exponential gradient environment, the numerical variations in m appear almost uniform over all x. To test the assumption A, we check the maximum of the normalized variation of cells methylation level: σ max m/mx 1 P + + P dm P + + P dm and also distinguish them by their moving directions: m/m ± σ ± x 1 P ± dm max P ±. dm, where M = ρ+ M + + ρ M ρ + + ρ, As shown in Figure, both σ and σ ± increases with G and decreases with k R, and they are much smaller than 1. i.e. Assumption A holds in these cases The localization of P ± in m in the limit of k R 1. We show by formal analysis that Assumption A is true when the adaptation rate k R 1. Denote 3.1 k R = 1/η, fa = f η a/η, then.1-. become P + P = v P + = v P 1 η 1 η f η ap + f η ap z P + P, + z P + P.

8 8 GUANGWEI SI, MIN TANG, AND XU YANG a Cells Up the Gradient Cells Down the Gradient M + M - G=.5µm -1 b Cells Up the Gradient Cells Down the Gradient M + M - G=.15µm Methylation level.6.4 Methylation level xµm xµm Figure 1. The distribution of cells receptor methylation level in exponential gradient environment [L] = [L] expgx. a G =.5µm 1 and b G =.15µm 1. The red dots represent cells moving up the gradient right side while the blue ones represent those moving down the gradient left side. M ± are the average methylation levels for the right and left moving cells respectively. In the simulation, we take [L] = 5K I. Other parameters are the same as those proposed in []. Integrating the above two equations with respect to m produces, for P ± R t, x = R P ± t, x, m dm R is an arbitrary positive constant, P + R P R = v P + R = v P R 1 R zp + P dm 1 η f η ar P + t, x, R + 1 η f η a P + t, x,, + 1 R zp + P dm 1 η f η ar P t, x, R + 1 η f η a P t, x,. The probability density functions satisfy P ± t, x, m, m, and thus P ± R t, x increases with R. We consider the regime 3.15 η 1, and f η a O1. Then when η 1, indicate for R, +, 3.16 f η ar P ± t, x, R = f η a P ± t, x, + Oη = Oη,

9 A MEAN-FIELD MODEL FOR CHEMOTAXIS 9 kr=.5s -1 G=.1µm σ σ + σ σ σ + σ - σ..1 σ G1-3 µm krs -1 Figure. The variances of cells methylation level for different G and k R. σ is defined as the maximum of normalized variation of m. σ ± are that of cells moving in + and - direction respectively. σ and σ ± increase with G for a given k R a and decrease with k R with fixed G b, and their values are much smaller than 1, as demanded by assumption A. where we have used the boundary condition that P ± t, x, m decays to zero at m =. Therefore, as η, 3.17 f η arp ± t, x, R, R, +. Then the definition of fa in.3-.4 gives that if R M, P ± t, x, R, which implies when η, 3.18 P ± x, t, m = ρ ± x, tδm M a, where, M a is defined by a[l]x, t, M a x, t = a, which makes fa =. Remark. When t P ± R, xp ± R are O1, the locally concentrated property depends only on how large η is, not the magnitude of z. Therefore, the assumption that z is large in the derivation of parabolic and hyperbolic scaling in the subsequent section will not effect the locally concentrated property here. In the large gradient environment or the chemical signal changes too fast, t P ± R, xp ± R become large and the locally concentrated assumption is no longer true. 4. Connections to the original PBMFT and the Keller-Segel limit In this section, we connect the new moment system to the original PBMFT developed in [7] from by taking into account the different physical time scales of the tumbling, adaptation and experimental observations. Especially, one of the equations delivers the important physical assumption eqn. 3 in [7]. We shall also derive the Keller-Segel limit when the system time scale is longer. Moreover,

10 1 GUANGWEI SI, MIN TANG, AND XU YANG a numerical comparison of the moment system with SPECS is provided in the environment of spatial-temporally varying concentration. We nondimensionalize the system by letting t = T t, x = L x, v = s ṽ, where T, L are temporal and spatial scales of the system respectively. Then J ρ = s Jρ, J q = s Jq, and the system becomes after dropping the 1 ρ + ρ + s T L 1 Z + ρ + Z ρ, T 1 1 ρ ρ s = v T L + 1 Z + ρ + Z ρ, T 1 1 q + q + s T L + 1 F + ρ + 1 M + Z + ρ + M Z ρ, T T 1 1 q q s = v T L + 1 F ρ + 1 M + Z + ρ + M Z ρ. T T 1 where T 1, T are the average run and adaptation time scales respectively. For E. coli, the average run time is at the order of 1s, the adaptation time is approximately 1s 1s, and according to the experiment in [36], the system time scale when the PBMFT can be applied is all those scales longer than 8s, while the Keller-Segel equation is only valid when the system time scale is longer than 1s. Therefore, for the PBMFT, we can consider the kinetic system under the scaling that the so-called hyperbolic scaling 4.1 T 1 L/s = ε, T L/s = 1, and T L/s = 1 with ε very small. On the other hand, for the Keller-Segel equation in the longer time regime, we consider the parabolic scaling such that 4. T 1 L/s = ε, T L/s = 1, and T L/s = 1 ε. In the subsequent part, when ε, we consider the following Hilbert expansions ρ ± = ρ ± + ερ ±1 +, q ± = q ± + εq ±1 +, 4.3 M ± = M ± + εm ±1 +, F ± = F ± + εf ±1 +, Z ± = Z ± + εz ±1 +. and use asymptotic analysis to connect to both PBMFT and Keller-Segel equation.

11 A MEAN-FIELD MODEL FOR CHEMOTAXIS The original PBMFT by the hyperbolic scaling. The macroscopic quantities in the PBMFT in [7] are the total density ρ s, the cell flux J s, the average methylation M s and the methylation difference M s. M +, M are the average methylation levels to the right and to the left. The connections of 3.1, 3. to the macroscopic quantities in PBMFT are: 4.4 ρ s = ρ + + ρ, J s = v ρ + ρ ; M s = 1 M + M = 1 The model in [7] is q + ρ s = J s, J s = v Z 1 s M s ρ + q ρ, together with the physical assumption ρ s v Zs 1 Ms = M + ρ + + M ρ ρ + + ρ Z s M sρ s F s J s ρ s M s 1 ρ s v M s ρ s, = q+ + q ρ + + ρ. 4.8 M s M s Z 1 s v, which physically means M s is approximated by the methylation level difference in the mean methylation field M s x, t over the average run length v Z 1, due to the fact that the direction of motion is randomized during each tumble event. Here Z s = zm s, F s = fm s. Under the scaling 4.1, become ρ + ρ Z + ρ + Z ρ, ε ρ ρ = v Z + ρ + Z ρ, ε q + q F + ρ + 1 ε q q 4.1 = v + F ρ + 1 ε M + Z + ρ + M Z ρ, M + Z + ρ + M Z ρ. Introducing the asymptotic expansions as in 4.3 and we first look at those leading order terms. Matching the O1/ε terms in 4.1 and 4.1 gives Z + ρ + = Z ρ, and M + Z + ρ + = M + Z ρ, which implies M + = M. Since za, fa are continuous function of m, Z + = Z, F + = F. Then Z + ρ + = Z ρ indicates ρ + = ρ and q + = M + ρ + =

12 1 GUANGWEI SI, MIN TANG, AND XU YANG M ρ = q. For simplicity, in the following part, we denote ρ = ρ ±, M = M ±, q = q ±, 4.13 Z = Z ±, F = F ± Z, = z. m=m ± On the other hand, let ρ s = ρ s + ερ 1 s +, J s = J s + εj 1 s + M s = M s + ε M 1 s +, M s = M s + εm 1 s +. Then the connections of the macroscopic quantities give 4.14 ρ s = ρ, J s =, M s =, M s = M ; ρ 1 s = ρ +1 + ρ 1, J 1 s = v ρ +1 ρ 1, M 1 s = 1 M +1 M 1. Moreover, it is important to note that, we have dropped the in the rescaled system , therefore, Z = zm s in is 4.15 Z s = zm + εm s 1 + = zm + Oε = zm + Oε = Z ε ε + O1. In the following part, we derive by asymptotics: Adding 4.9 and 4.1 brings 4.5. Subtracting 4.9 by 4.1 gives J s = ρ s v v Z + ρ + Z ρ ε Since Z ± = zm ±, [L] = z M ± + εm ±1 +, [L] = zm, [L] + ε z M ±1 + = Z + ε Z m=m M ±1 +, we find 4.16 Z +1 Z 1 = Z M +1 M 1. Then O1 terms of subtracting 4.9 by 4.1 yield 4.17 J s = v = v = v ρ s v Z + ρ +1 Z ρ 1 v Z +1 ρ + Z 1 ρ ρ s Z v ρ +1 ρ 1 Z ρ v M +1 M 1 ρ s Z J 1 s ρ ρ s Z s v M 1 s = v Z Z s sj s v M sρ s + Oε Here in the first equation, we have used In the last two equation, we have used 4.14, 4.15 and it is accurate to Oε.

13 4.18 A MEAN-FIELD MODEL FOR CHEMOTAXIS 13 Then from J s =, v ρ s and we get 4.6. Adding 4.11 and 4.1 gives q + + q From 4.14, 4.4, we have Z sj s v Z s M sρ s = q + q v q + q = v ρ + M + ρ M and + F + ρ + + F ρ. = εm v ρ +1 ρ 1 + ερ v M +1 M 1 + Oε = M s J s + ρ s v M s + Oε, F + ρ + + F ρ = fm + ρ + + fm ρ = f M s + M + M s ρ + + f M s + M M s ρ = F s ρ s + f M + M s ρ + + M M s ρ + Oε m=ms = F s ρ s + Oε. Therefore, 4.18 is equivalent to M s ρ s ρ s M s ρ s M s M s J s M s J s J s + F s ρ s + Oε M s + F sρ s + Oε. By using 4.5, the above equation is the same as 4.7 and it is accurate up to Oε Finally, from 4.13, the O1 terms of subtracting 4.11 by 4.1 yield q v Z M +1 M 1 +M Z +1 Z 1 ρ M Z ρ +1 ρ 1 = Noting the first equation in 4.17, the above equation is equivalent to M v 1 J s v ρ M Z M 1 s ρ = Thanks to J s =, M s = from 4.14 and the relation of Z and Z in 4.15, the above equation leads to the important physical assumption 4.8, 4.19 M s M Z 1 v, We have recovered the PBMFT model in [7].

14 14 GUANGWEI SI, MIN TANG, AND XU YANG Remark. In the derivation of the PBMFT, we have decomposed the tumbling frequency into two different scales. This idea is similar to the general derivation approach in [17], but we have additional equations for the time evolution of the methylation level. Since the turning operator depends on the methylation level which also changes dynamically, it is hard to determine explicitly how the turning operator depends on [L] as in [17]. According to [17], the Hilbert approach indicates that the PBMFT is an approximation of order ε of the transport system , which is not clear for the moment system in [15, 16, 35]. 4.. Keller-Segel limit by the parabolic scaling. Under the scaling 4., become ε ρ+ ε ρ ε q+ ε q ρ + 1 ε ρ = v + 1 ε q + Z + ρ + Z ρ, Z + ρ + Z ρ, + F + ρ + 1 ε q = v + F ρ + 1 ε M + Z + ρ + M Z ρ, M + Z + ρ + M Z ρ. First of all, we have similar equations as in Besides, the O1 terms in and 4.13 yield F ± =, which is the main difference between the hyperbolic and parabolic scaling. Then equating the Oε terms in adding 4. and 4.1 together produces 4.4 ρ = v ρ +1 ρ 1. Putting together the O1 terms in subtracting 4. by 4.1 and subtracting 4. by 4.3 brings ρ v Z ρ +1 ρ 1 ρ Z +1 Z 1 =, q + + q v M Z ρ +1 ρ 1 Z M +1 M 1 + M Z +1 Z 1 ρ =. Multiplying 4.5 by M and subtracting it from 4.6 give v ρ M Z ρ M +1 M 1 =.

15 A MEAN-FIELD MODEL FOR CHEMOTAXIS 15 Then, from 4.16, the two equations 4.5 and 4.6 imply ρ +1 ρ 1 = Z 1 ρ v Z M +1 M 1 ρ 4.7 = Z 1 ρ v + v Z 1 Z M ρ Z 1 ρ + v Z Substituting 4.7 into 4.4 gives the K-S equation ρ 4.8 = v Z 1 ρ v Z Z M ρ Using M = M a, Z = zm a, the latter equation becomes 4.9 with χ = ρ = v Z 1 v τ 1 z + τ 1 NH1 a. Remark. 1. If instead of 4., we consider T 1 L/s = ε, then the rescaled system becomes ε ρ+ ε ρ ε q+ ε q ρ + 1 ε ρ = v + 1 ε q + T L/s = κε, ρ and Z + ρ + Z ρ, Z + ρ + Z ρ, + 1 κε F + ρ + 1 ε q = v + 1 κε F ρ + 1 ε Z M ρ. χ ρ f T L/s = 1 ε, M + Z + ρ + M Z ρ, M + Z + ρ + M Z ρ. When κ O1/ε, carrying on similar asymptotic expansion will produce the same Keller-Segel limit 4.9 as ϵ. This indicates that when the adaptation time is shorter than T T 1, the Keller-Segel equation is valid for E. coli chemotaxis.. The velocity scale of individual bacteria is s. The temporal and spacial scales of the system we consider are T and L respectively, therefore the velocity scale of the drift velocity v d = J ρ /ρ is L/T. The scaling 4. implies v d /s Oε, which means that in the regime where K-S equation is valid, the drift velocity is much smaller than the moving velocity of individual bacteria. 3. If the adaptation is faster than the characteristic tumbling time, which indicates that E. coli can adapt to the environment almost immediately, it exhibits no chemotactic behavior since the tumbling frequencies are the same in moving different directions.

16 16 GUANGWEI SI, MIN TANG, AND XU YANG u=.4µm/s u=8µm/s ρx 1-3 ρx 1-3 a b c d 3.3 SPECS Ma PBMFT xµm xµm xµm xµm e f g h SPECS Ma.1 PBMFT J ρ Jρ M M Jq Jq xµm xµm xµm xµm Figure 3. Numerical comparison between the new transport system and SPECS. The steady state profiles of ρ: a, e, J ρ : b, f, M = q/ρ: c, g, J q : d, h when the traveling wave speeds are u = 8µm/s and u =.4µm/s respectively. In the subfigures, red lines, histograms and dots are from SPECS red lines in a and e are the estimated probability densities of the red histograms; red lines in b, d, f, h are the smoothed results of the red dots, while blue lines are calculated using the new transport system of PBMFT. Parameters used here are [L] = 5µM, [L] A = 1µM, λ = 8µm. cells are simulated in SPECS Numerical comparison in the traveling wave concentration. To show the validity of the moment system , numerical comparisons to SPECS will be presented in this subsection. We choose spatial-temporal varying environment to show how the intracellular dynamics affects the E. coli behaviors at the population level. We consider a periodic 1-D domain with the traveling wave concentration given by [L]x, t = [L] +[L] A +sin[ π λ x ut]. The wavelength λ is fixed to be the length of the domain, while the wave velocity u can be tuned. The traveling wave profiles of all the macroscopic quantities in and corresponding SPECS results are compared in Figure 3. The results from SPECS and the moment system are quantitatively consistent. It can be noticed that, when the concentration changes slowly u =.4µm/s, the profile of M can catch up with the target value M a defined by a[l], M a = a, while in the fast-varying environment u = 8µm/s there is a lag in phase between M and M a. This difference is caused by the slow adaptation rate of cell and it also leads to the difference in the profiles of ρ and even chemotactic velocity; we refer interested readers to [7] for more detailed discussions and physical explanations.

17 A MEAN-FIELD MODEL FOR CHEMOTAXIS Two dimensional mean-field model In this section, we derive the two-dimensional moment system of PBMFT based on a formal argument using the point-mass assumption in methylation and the minimization principle proposed in []. In two dimensions, v = v cos θ, sin θ, where v is the velocity magnitude. P t, x, v, m in.7 can be rewritten as P t, x, θ, m. zm, [L], θ, θ is the tumbling rate from θ to θ. The tumbling term QP, z in.8 becomes 5.1 QP, z = zm, [L], θ, θ P t, x, θ, m dθ zm, [L], θ, θ dθ P t, x, θ, m, V where V = [, π and = 1 π V. According to.6, zm, [L], θ, θ is independent of θ and thus we denote it by zm, [L]. Define 5. gt, x, θ = P t, x, θ, m dm, ht, x, θ = mp t, x, θ, m dm; V 5.3 Mt, x, θ = ht, x, θ gt, x, θ and the density and momentum in m as follows ρ f t, x = gt, x, θ dθ, ρ b t, x = vgt, x, θ dθ; V f V b ρ u t, x = gt, x, θ dθ, ρ d t, x = vgt, x, θ dθ; V 5.4 u V d q f t, x = ht, x, θ dθ, q b t, x = vht, x, θ dθ; V f V b q u t, x = ht, x, θ dθ, V u q d t, x = vht, x, θ dθ; V d where = π and V f = 7π/4, [, π/4, V b = 3π/4, 5π/4, V u = π/4, 3π/4, V d = 5π/4, 7π/4. We assume 5.5 P t, x, θ, m = gt, x, θδ m Mt, x, θ. This assumption is motivated by 3.18 in one dimension, which could be formally understood as the limit of k R +. Let i represent all four possible superscript f, b, u, d, denote 5.6 M i = qi ρ i, Zi = zm i, Z i = z M i, F i = fm i, F i = f M i.

18 18 GUANGWEI SI, MIN TANG, AND XU YANG Integrating.7 with respect to m yields 5.7 t g = v x g + z Mθ, [L] gt, x, θ dθ z Mθ, [L] gt, x, θ. V Integrating 5.7 with respect to θ from 7π/4 to π and to π/4 gives the equation for ρ f, 5.8 ρ f t, x v x g dθ 3 Z f + Zf V f 4 V f M M f g dθ + 1 Z b + Zb 4 V b M M b g dθ + 1 Z u + Zu 4 V u M M u g dθ + 1 Z d + Zf 4 V d M M d g dθ = v x g dθ 3 V f 4 Zf ρ f Zb ρ b Zu ρ u Zd ρ d Similar equations can be found for ρ b, ρ u, ρ d but we exchange the superscript f with b, with u and with d respectively. Multiplying.7 by m and integrating it with respect to m bring 5.9 t h = v x h + f Mθ, [L] gθ + V z Mθ, [L] gθ Mθ dθ zmθ, [L]gθMθ. Integrating 5.9 with respect to θ from 7π/4 to π and to π/4, and using the definition in 5.3 give 5.1 q f x, t v x h dθ + F f + F f V f V f M M f gx, t, θ dθ 3 Z f M f + Z f + M f Zf M M f g dθ 4 V f + 1 Z b M b + Z b + M b Zb M M b g dθ 4 V b + 1 Z u M u + Z u + M u Zu M M u g dθ 4 V u + 1 Z d M d + Z d + M d Zd M M d g dθ 4 V d = v x h dθ + F f ρ f 3 V f 4 Zf q f Zb q b Zu q u Zd q d Similar equations can be found for q b, q u, q d but we exchange the superscript f with b, with u and with d respectively.

19 A MEAN-FIELD MODEL FOR CHEMOTAXIS 19 In order to close the system, we need a constitutive relation that represents V i v x g dθ and V i v x h dθ by ρ i, q i i represents f, b, u, d. Assume 5.11 gt, x, θ g 1 t, x + π g ct, x cos θ + π g st, x sin θ + π g ct, x cos θ, ht, x, θ h 1 t, x + π h ct, x cos θ + π h st, x sin θ + π h ct, x cos θ. Then from 5.4, ρ f t, x q f t, x gt, x, θ dθ = g 1 + g c + g c, V f ht, x, θ dθ = h 1 + h c + h c, V f Similarly ρ b g 1 g c + g c, ρ u g 1 + g s g c, ρ d = g 1 g s g c ; q b h 1 h c + h c, q u h 1 + h s h c, q d = h 1 h s h c. Therefore, expressing g 1, g c, g s, g c, h 1, h c, h s, h c by ρ f, ρ b, ρ u, ρ d, q f, q b, q u, q d, we find g 1 = 1 ρ f + ρ b + ρ u + ρ d, g c = ρ f + ρ b ρ u ρ d, 5.1 g c = ρ f ρ b, g s = ρ u ρ d, 4 4 h 1 = 1 q f + q b + q u + q d, h c = 1 q f + q b q u q d, 4 4 h c = 4 q f q b, h s = q u q d, 4 Hence, 5.13 v x g dθ V f π xg 1 + π v x g dθ V b π v x g dθ V u π yg 1 + π 4 1 v x g dθ V d π g c + π 4 1 y g s + 3 xg c, xg 1 + π g c + π 4 1 y g s 3 xg c, g c + π y g s 3 yg c yg 1 + π 4 1 g c + π y g s + 3 yg c

20 GUANGWEI SI, MIN TANG, AND XU YANG 5.14 v x h dθ V f π xh 1 + π v x h dθ V b π v x h dθ V u π yh 1 + π 4 1 v x h dθ V d π h c + π 4 1 y h s + 3 xh c, xh 1 + π h c + π 4 1 y h s 3 xh c, h c + π y h s 3 yh c yh 1 + π 4 1 h c + π y h s + 3 yh c Furthermore, noting 5.6, we are able to close the system 5.8, 5.1 and those equations for ρ b, ρ u, ρ d and q b, q u, q d, using 5.1, 5.13, If, instead of 5.11, other dependence of g, h on θ is applied, different system can be obtained. In summary, we get an eight equation two-dimensional system that is similar to The main assumption made here is that the methylation level is locally concentrated in each direction, but it can vary in different directions, which gives direction dependent tumbling frequency, and thus chemotactic behavior. The eight equation system we obtained can be considered as a semi-discretization in the velocity space of the original two dimensional system.7. We can derive a similar PBMFT system as in by asymptotics. 6. Discussion and conclusion To seek a model at the population level that incorporates intracellular pathway dynamics, we derive a new kinetic system in this paper under the assumption that the methylation level is locally concentrated. We show via asymptotic analysis that, the hydrodynamic limit of the new system recovers the original model in [7]. Especially, the quasi-static approximation on the density flux and the assumption on the methylation difference made in [7] can be understood explicitly. We show that when the average run time is much shorter than that of the population dynamics parabolic scaling, the Keller-Segel model can be achieved. Some numerical evidence is shown to present the quantitative agreement of the moment system with SPECS []. We remark that the idea of incorporating the underlying signaling dynamics into the classical population level chemotactic description has appeared in the pioneering works of Othmer et al [15, 16, 35]. The model of the internal pathway dynamics used here are based on quantitative measurement by in vivo FRET experiments and proposed recently. An open question related to the chemo-sensory system of bacteria still remains in the large gradient environment, in which the distribution of the methylation level is no longer locally concentrated. It would be interesting to study and improve the macroscopic model in large gradient environment.

21 A MEAN-FIELD MODEL FOR CHEMOTAXIS 1 References [1] J. Adler, Chemotaxis in bacteria, Science , [] U. Alon, M.G. Surette, N. Barkai, and S. Leibler, Robustness in bacterial chemotaxis, Nature , [3] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol , [4] H.C. Berg, Motile behavior of bacteria, Physics Today 53, 4 9. [5] H.C. Berg and D.A. Brown, Chemotaxis in Escherichia coli analysed by three-dimensional tracking, Nature , [6] P. Biler, L. Corrias, and J. Dolbeault, Large mass self-similar solutions of the parabolic Keller-Segel model of chemotaxis, J. Math. Biol , 1 3. [7] A. Blanchet, J.A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical patlakkeller-segel model in R, Comm. Pure Appl. Math. 61 8, [8] A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 44 6, 1 3. [9] N. Bournaveas and V. Calvez, Global existence for the kinetic chemotaxis model without point wise memory effects, and including internal variables, Kinetic and Related Models 1 8, [1] D. Bray, M.D. Levin, and K. Lipkow, The chemotactic behavior of computer-based surrogate bacteria, Curr. Biol. 17 7, [11] A. Celani and M. Vergassola, Bacterial strategies for chemotaxis response, Proc. Natl. Acad. Sci. 17 1, [1] F.A.C.C. Chalub, P.A. Markowich, B. Perthame, and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh.Math. 14 4, [13] P. Cluzel, M. Surette, and S. Leibler, An ultrasensitive bacterial motor revealed by monitoring signalling proteins in single cells, Science 87, [14] R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, Discrete and Continuous Dynamical Systems Series B 6 6, [15] R. Erban and H.G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math. 65 4, [16], From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model. Simul. 3 5, [17] F. Filbet, P. Laurencot, and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol. 5 5, [18] G. L. Hazelbauer, J. J. Falke, and J. S. Parkinson, Bacterial chemoreceptors: highperformance signaling in networked arrays, Trends Biochem. Sci. 33 8, [19] G. L. Hazelbauer and S. Harayama, Sensory transduction in bacterial chemotaxis, Int. Rev. Cytol , [] T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models and Methods in Appl. Sci. 1, [1] T. Hillen and H.G. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math. 61, [] L. Jiang, Q. Ouyang, and Y. Tu, Quantitative modeling of Escherichia coli chemotactic motion in environments varying in space and time, PLoS Comput. Biol. 6 1, e1735. [3] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol , 5 34.

22 GUANGWEI SI, MIN TANG, AND XU YANG [4] J.E. Keymer, R.G. Endres, M. Skoge, Y. Meir, and N.S. Wingreen, Chemosensing in escherichia coli: two regimes of two-state receptors, Proc. Natl. Acad. Sci. U.S.A. 13 6, no. 6, [5] B.A. Mello and Y. Tu, Quantitative modeling of sensitivity in bacterial chemotaxis: The role of coupling among different chemoreceptor species, Proc. Natl. Acad. Sci. 1 3, [6] T.S. Shimizu, Y. Tu, and H.C. Berg, A modular gradient-sensing network for chemotaxis in Escherichia coli revealed by responses to time-varying stimuli, Mol. Syst. Biol. 6 1, 38. [7] G. Si, T. Wu, Q. Ouyang, and Y. Tu, A pathway-based mean-field model for Escherichia coli chemotaxis, Phys. Rev. Lett. 19 1, [8] J.E. Simons and P.A. Milewski, The volcano effect in bacterial chemotaxis, Math. Comput. Model , [9] V. Sourjik and H.C. Berg, Receptor sensitivity in bacterial chemotaxis, Proc. Natl. Acad. Sci. 99, [3] A. Stevens, Derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems, SIAM J. Appl. Math. 61, [31] M.J. Tindall, P.K. Maini, S.L. Porter, and J.P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations, Bull. Math. Biol. 7 8, [3] M.J. Tindall, S.L. Porter, P.K. Maini, G. Gaglia, and J.P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell, Bull. Math. Biol. 7 8, [33] Y. Tu, T.S. Shimizu, and H.C. Berg, Modeling the chemotactic response of escherichia coli to time-varying stimuli, Proc. Natl. Acad. Sci. U.S.A. 15 8, no. 39, [34] G. Wadhams and J. Armitage, Making sense of it all: bacterial chemotaxis, Nat. Rev. Mol. Cell Biol. 5 4, [35] C. Xue and H.G. Othmer, Multiscale models of taxis-driven patterning in bacterial pupulaitons, SIAM J. Appl. Math. 7 9, [36] X. Zhu, G. Si, N. Deng, Q. Ouyang, T. Wu, Z. He, L. Jiang, C. Luo, and Y. Tu, Frequencydependent Escherichia coli chemotaxis behavior, Phys. Rev. Lett. 18 1, Center for Quantitative Biology, Peking University, Beijing, China, 1871, gwsi@pku.edu.cn Institute of Natural Sciences, Department of mathematics and MOE-LSC, Shanghai Jiao Tong University, 4, Shanghai, China, tangmin@sjtu.edu.cn Department of Mathematics, University of California, Santa Barbara, CA 9316, xuyang@math.ucsb.edu