Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

Transcription

1 Journal of Mathematical Biology, accepted copy Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling Chuan Xue Received: date / Revised: date Abstract Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak- Keller-Segel (PKS) equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived s and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy. Keywords Multiscale Analysis Bacterial Chemotaxis Cell Signaling Keller-Segel Velocity Jump Process Mathematics Subject Classification () 9B5 9C7 9D5 35Q A J75 CX is supported by the National Science Foundation in the United States through grant DMS-39. CX is also supported by the Mathematical Biosciences Institute at the Ohio State University as a long-term visitor. Chuan Xue Department of Mathematics, Ohio State University, Columbus, OH 3. Tel.: cxue@math.osu.edu

2 Chuan Xue Introduction Chemotaxis is the active movement of cells or organisms in response to external chemical signals. It is crucial in many multicellular processes such as biofilm-associated infections, bioremediation, embryonic development, wound healing and cancer metastasis [,,, 9,, 9, 5,,, 7]. To develop quantitative and predictive models of these processes, a crucial step is to accurately describe chemotaxis of cell populations. Chemotaxis of cell populations has been modeled using both continuum and cell-based approaches. Continuum models of chemotaxis use partial differential equations (s) to describe the evolution of the cell density. Among different models, the most popular is the classical Patlak-Keller-Segel (PKS) chemotaxis equation (or its variant forms), n t = D n n χn S, (.) where n = n(x, t) is the cell density, S = S(x, t) is the signal concentration, D n is the diffusion coefficient, χ = χ(s, S, n,...) is the chemotactic sensitivity, and u S = χ S is the macroscopic chemotactic velocity. The PKS equation was first introduced in [5], and later applied to model chemotaxis of bacteria and amoeboid cells in [3 3]. Following these pioneering work, the PKS equation has been applied to describe chemotaxis of all kinds of cells and also cell movement to signals other than chemical signals. In these models, the forms of D n and χ are often chosen phenomenologically. Mathematical properties of these models have been studied extensively (see [5, 7, 7] for recent reviews). However, fundamental questions such as under what conditions Equation (.) can be justified and how to relate D n and χ to the molecular mechanisms of chemotaxis are still purely understood. On the other hand, cell-based models of chemotaxis have been proposed to treat cells as interacting particles and simulate the evolution of each cell s intracellular state and trajectory. These models can incorporate many details of cellular processes such as signal transduction and cell movement, and have been used to address population dynamics and pattern formation [, 5, 9, 35, 75]. However for applications that involve large numbers of cells ( and above), computation of these models is extremely expensive and parameter exploration extremely time-consuming. Therefore to efficiently and accurately model chemotaxis at the population level, it is crucial to establish quantitative connections between these two distinct approaches. The scope of this paper is to address these issues for bacterial chemotaxis. Chemotaxis of run-and-tumble bacteria has been extensively studied over the past 5 years. Examples of such bacteria include Escherichia, Salmonella, Bacillus, Rhodobacter, and Pseudomonas [3,, 3, 5]. Among them the best understood is the model system Escherichia coli [ 7, 3, 37]. These bacteria move by alternating forward-moving runs and reorienting tumbles. In the absence of signal gradients, the overall cell movement is an unbiased random walk. However, when exposed to a signal gradient, cells bias their movement towards higher (lower) concentrations of the signal if it is an attractant (repellent). This modification of movement is orchestrated by a very complicated chemotaxis signal transduction pathway, which involves a rapid response of the cell to the external signal change termed excitation, and a slow adaptation which allows the cell to subtract out the background signal and respond to further signal changes. Recently, some progress has been made in deriving macroscopic s from cartoon cell-based models of bacterial chemotaxis [3, 3]. In dilute solutions, the run-and-tumble movements of these cells can be described as independent velocity jump processes with a bias towards higher attractant concentrations or lower repellent concentrations []. In [, 5, 57], velocity jump models were analyzed and Equation (.) was derived as an

3 Macroscopic equations for bacterial chemotaxis 3 approximation of the macroscopic cell density in the diffusion limit. In these works, the parameters of the velocity jumps were assumed to depend on the external signal directly, and intracellular signaling was not included. To understand how to embed information of intracellular signaling into population dynamics, Erban and Othmer [7, ] considered a cell-based model which incorporates a linear cartoon description of the excitation and adaptation response of a cell and determines the rate of velocity jumps as a linear function of the output of the cartoon model. Equation (.) was derived under a shallow gradient assumption to approximate the macroscopic cell density. In [7], these analysis were extended to allow nonlinear, more realistic velocity jump rates and external forces acting on cells. The works in [7,,7] focused on the development of mathematical methods to derive the PKS equation (.) from cell-based models of bacterial chemotaxis. The linear cartoon model, chosen for its simplicity, is a conceptual model and not based on real intracellular signal transduction pathways. Due to its abstractness and simplicity, these works could not address the following significant questions: First, how do nonlinearities in intracellular signaling affect the overall strategy of population dispersal? Second, how do the diffusion rate and chemotactic sensitivity quantitatively depend on the structure and kinetics of the intracellular signaling network? Third, it was shown in [75] that the PKS equations derived in [7,, 7] are limited to chemotactic responses to signals with shallow gradients, but how to interpret this mathematically speaking shallow gradient assumption in terms of biological data and to what extent do the derived equations provide valid approximations of the underlying cell-based models when this assumption is violated? To address these questions, macroscopic equations and quantities have to be derived from cell-based models that incorporate real intracellular signaling networks which are intrinsically nonlinear and much more complicated. In this paper, we address these issues by considering cell-based models of bacterial chemotaxis based on detailed biochemistry of intracellular signaling. First, starting from a general formulation of the cell-based model, we analytically show that under a small signal variation assumption, the dynamics of the macroscopic cell density can be approximated quantitatively by Equation (.) in arbitrary space dimensions. Furthermore, we give explicit formulas for the flux coefficients D n and χ in terms of the general description for intracellular signaling (Equations (3.) and (3.3), or (3.9) for a special case). These formulas suggest that nonlinearities of intracellular signaling affect the population dynamics through χ. Given a specific bacterium or other individual that move with a similar strategy, these formulas can be readily applied to extract population-level properties as long as there is a detailed description of single cell signaling and movement. Second, we apply these general formulas to chemotaxis of the model bacterium E. coli, and use them to draw connections between properties of single cell signaling and population dynamics. Our theory suggests that the structure of the adaptation network can explain the logarithmic sensing property, i.e., the macroscopic drift of the population is proportional to the gradient of the logarithm of the signal [3]. This provides a detailed mechanistic basis for the logarithmic sensitivity of E. coli chemotaxis. This application illustrates the usefulness of the general formulas in elucidating connections between microscopic dynamics and macroscopic behavior in bacterial chemotaxis. Third, we numerically investigate the scope of applicability of the derived equations and formulas, i.e., whether the derived equations and formulas provide good approximations of the underlying cell-based models when the small signal variation assumption is violated. Using E. coli chemotaxis as an example, our simulations show that the derived equations and formulas match the underlying cell-based models quantitatively even if the small signal variation assumption is not satisfied, as long as the internal states of most cells are close to the adapted states. When the signal change becomes extremely fast, most cells are far from

4 Chuan Xue their adapted states and the derived equation at most qualitatively agrees with the underlying cell-based model. To further determine whether these situations occur in biological settings such as [,,, 59, 7] requires joint investigation of experimentalists and theoreticians. The organization of the paper is as follows. In Section, we first introduce the biology for bacterial chemotaxis and the general framework of cell-based models of it. In Section 3 we summarize our main analytical and simulation results in biological terms, and these results will be presented in detail in Sections and 5. In Section we derive Equation (.) from cell-based models under the small signal variation assumption and give general formulas of D n and χ. This section is intended for a more theoretical audience, and can be skipped by a reader who is not interested in the mathematical details without affecting the understanding of the rest of the paper. In Section 5, we apply the general theory to chemotaxis of E. coli. We use three cell-based models with increasing complexity. The first uses the linear cartoon model of cell signaling considered in [7,,7]. We show that the results of this paper are consistent with previous work and the latter can be thought as a special case of the former. The second uses a coarse-grained nonlinear model of E. coli signaling which is based on the main structure of the signaling network, and the third is based on a very comprehensive model of E. coli signaling. Both cases recover the logarithmic sensing property. In Section 5 we also illustrate the scope of applicability of the derived equations and formulas when the small signal variation assumption is violated. Finally, in Section, we discuss our results and identify future directions. Bacterial chemotaxis and cell-based models. Biology of bacterial chemotaxis Bacteria can swim using a run and tumble strategy by rotating their flagella [3,, 5]. When rotated counterclockwise (CCW) the flagella form a bundle and the cell is propelled forward with a speed 3 µm/s; when rotated clockwise (CW) the bundle flies apart and the cell tumbles in place. After a tumble, the cell picks a new direction randomly and runs again. In the absence of signal gradients, the overall cell movement is an unbiased random walk, with a mean run time s and a mean tumble time. s. However, when exposed to a signal gradient, the cell increases (decreases) the mean run time when moving towards (away from) a favorable direction, and thus has an overall drift towards better environments [7]. The details of movement for different bacteria can be slightly different. For example, for E. coli, the new direction after a turn has a slight bias in the previous direction and the turn angle has a unimodal distribution [7], while for Pseudomonas, the turn angle shows a bimodal distribution []. The change of swimming pattern in response to chemical signals is orchestrated by a complicated signal transduction pathway, which has been studied extensively in the past few decades [3,7]. Figure shows the key components of the signal transduction pathway for E. coli chemotaxis. The transmembrane chemoreceptors form stable ternary complexes with the signaling proteins CheA and CheW, and cluster at one pole of the cell body. CheA is an auto-kinase, and its kinase activity is reduced if attractants bind to the associated receptor. CheA is also a kinase for the response regulators CheY and CheB. The phosphorylated form CheY p binds to the flagella motor, increases the probability of CW rotation, and thus triggers tumbling. On the other hand, CheB p and CheR change the methylation state of the receptor at a slower rate: CheR methylates it and CheB p demethylates it. This methylationdemethylation cycle restores the activity of the associated CheA. In summary, upon ligand

5 Macroscopic equations for bacterial chemotaxis 5 Fig. The chemotaxis signaling pathway for the model bacterium E. coli. Transmembrane chemoreceptors function as trimers of dimers with ligand-binding domains on the periplasmic side and signaling domains on the cytoplasmic side. Methylation sites of receptors appear as white dots on the receptors. The cytoplasmic signaling proteins are represented by single letters, e.g., A = CheA. Red (blue) components promote CCW (CW) rotation of flagellar motors. Reprinted from [7] with permission. binding, the kinase activity of CheA is reduced, thus CheY p decreases rapidly, and the cell tends to run for longer. This process, called excitation, occurs within fractions of seconds. Simutaneously, CheB p is reduced but CheR is not affected, thus the receptor methylation level increases, until the activity of CheA is restored to its pre-stimulus level. This process, called adaptation, takes from seconds to minutes, depending on the nature of the signal. We note that excitation and adaptation are two concurrent processes that affect each other, specifically, adaptation acts as a negative feedback to excitation which is directly induced by external signal changes. It is crucial for cells to subtract out background signal and respond to further signal changes. E. coli chemotaxis system is well-known for its ultra sensitivity and robust adaptation: cells can detect signals that span over 5 orders in concentration and respond to a change in the occupancy of the receptor as little as.% []. Recent experiments show that E. coli cells respond to the logarithm of the external signal in the sense that the macroscopic drift of the cell population, u S, increases with the gradient of the logarithm of the attractant, i.e., log(s + α ) where α is a positive constant [3]. In these experiments, the value of log(s + α ) is relatively small and the observed macroscopic drift u S is much smaller than the cell speed. Other bacteria species use similar strategies to move chemotactically as E. coli, but their signal transduction pathway can be substantially different. For example, the signal transduction pathway for Bacillus subtilis includes components CheC, CheD, and CheV which are not found in E. coli, and has three adaptation mechanisms instead of one [55,5]. Another example is Rhodobacter sphaeroides. Unlike E. coli, it has multiple homologs of the signaling proteins CheB and CheY which respond to two clusters of receptors [53]. Despite these differences, the robustness and effectiveness of the sensing mechanism seem to be ubiquitous across species [3].

6 Chuan Xue. Cell-based models of bacterial chemotaxis Significant effort has been put into modeling the intracellular signaling of bacterial chemotaxis quantitatively and this has greatly advanced our understanding of the molecular mechanism of bacterial chemotaxis [7,,5]. The detailed biochemistry of intracellular signaling is usually modeled as a system of ODEs which fall within the following general form dy dt = f`y, S(x, t), (.) Here y R q is the intracellular protein concentrations with q being the dimension of the ODE system, x R N is the cell position, and S(x, t) is the extracellular signal along the cell trajectory. The physiological domain of y, denoted as Y, is typically a compact set in the positive cone of R q. The signal S(x, t) satisfies S(x, t) [, S max] where the constant S max is the upper bound of the signal that a cell can possibly respond to. In the rest of the paper, we omit the dependence of S on x and t for simplicity of notation. The run-and-tumble movement of swimming bacteria can be modeled as a velocity-jump process. A velocity jump process is a process in which the velocity of an individual jumps instantaneously according to a turning rate λ and a turning kernel T (v, v ) []. For bacterial chemotaxis, the turning rate specifies the frequency of reorienting tumbles, and depends on the external signal S indirectly through the intracellular variable y, i.e., λ = λ(y). The turning kernel T (v, v ) specifies the probability of direction transition from v to v during a tumble, and is assumed to be independent of y. We note that in order for T (v, v ) to be a probability kernel, it has to satisfy that Z T (v, v )dv =, (.) V reflecting the fact that a cell turning from v has to choose a new direction with probability. In the case of E. coli chemotaxis, the turning rate is a function of the concentration of the motor-binding protein CheYp, i.e., λ = λ(y p). In general, the equation for y p can either be included explicitly as a variable in the ODE system (.), or given implicitly as a function of slower-changing variables by a system of steady-state equations. In each case, we have y p = y p(y), and thus in general we have λ = λ(y). Let p(x, v, y, t) be the probability density for a cell to have position x R N, velocity v V R N, and internal states y R q at time t. Here V is the velocity space which consists of all possible velocities that a cell can take. For example, for a cell that swims with a constant speed s but can swim in any direction, V is the sphere centered at the origin with radius s. Based on the general theory in [5], p(x, v, y, t) satisfies the following transport equation p Z t + x vp + y f(y, S)p = λ(y)p + λ(y) T (v, v )p(x, v, y, t)dv. (.3) where f(y, S) is given by the right-hand side of (.). Here the x-divergence term describes the change of probability due to cell run, the y-divergence term is due to evolution of the internal states, and the right-hand side models the reorientation of the cell during tumble. When cells are sufficiently separated, cell-cell mechanical interactions can be neglected and cells move independently. Thus the above description can be applied to describe every single cell and simulate the evolution of a large population of cells. This type of model is V

7 Macroscopic equations for bacterial chemotaxis 7 usually called a cell-based model of bacterial chemotaxis. It is often coupled with continuum equations for the external chemical signals, which depend on the macroscopic cell density n(x, t). This approach is called hybrid modeling methods [9], and has been used to generate significant insights on the mechanisms of bacterial self-organization and pattern formation [, 75]. However, as the cell population grows, the computational cost of the cell-based component is high and a continuum model for bacterial chemotaxis is desired. We next derive approximating s for the macroscopic cell density n(x, t) from cell-based models of bacterial chemotaxis governed by the general master equation (.3). The main results and biological interpretations are summarized in Sec 3, and the analytical details are included in Sec. In Section 5, we take E. coli chemotaxis as an example, and apply our general theory to explain the origin of the logarithmic sensitivity of E. coli. 3 Macroscopic equations for bacterial chemotaxis This section is a summary of our theory developed in Sections and 5. If the external signal changes slowly, cell-based models of bacterial chemotaxis that fall under the general framework introduced in Section. can be approximated by continuum models that use (.) to describe the evolution of cell densities. We give explicit formulas for the diffusion coefficient D n and the chemotactic sensitivity χ in terms of detailed knowledge of individual cell signaling and movement. To our knowledge, this is the first general, explicit formula that connects the detailed intracellular signaling dynamics with the chemotactic sensitivity for bacterial chemotaxis in arbitrary space dimensions. The detailed mathematical analysis that leads to these results is included in Section and applications of the general theory to E. coli chemotaxis is included in Section Properties of the cell-based models We first state the properties of the cell-based models that we consider.. Since the system (.) describes the intracellular signal transduction of a bacterium in response to the external signal, the function f on the right-hand side of (.) consists of terms based on mass-action kinetics, Michaelis-Menten functions, Hill functions, or any other functions that are smooth and differentiable in the physiological range of the variable y.. Bacteria can effectively adapt to step signal changes with a large range of magnitudes, and the adaptation is quite robust. Due to these properties, it is expected that there is a unique adapted state ȳ(s) of the system (.), satisfying f(s, ȳ) =, and furthermore, given any fixed S = S, the steady state ȳ(s ) is globally asymptotically stable. This means that for any constant external signal S, no matter where the internal states y start at time, y always converge exponentially to the adapted state ȳ(s ) as time evolves. In the mathematical analysis in Section, we use a weaker version of this property, that is, for any constant S, all the eigenvalues of the Jacobian matrix ( yf) y=ȳ(s) have negative real parts. 3. We note that for models that are based on the bacterial signaling pathways, components of y represents protein concentrations, and thus ȳ(s) are nonnegative functions of S. Components of ȳ representing chemoreceptors, CheR and CheB concentrations typically depend on the signal S, and components of ȳ representing CheYp and CheZp are typically independent of S or only weakly depend on S.

8 Chuan Xue. The cell s basal turning rate, given by λ = λ(ȳ(s)), is positive. This means that cells that are fully adapted to the external signal tumble at random time points instead of move in one direction persistently. In the case of E. coli, λ has been estimated to be s [5]. 5. When a cell stops, it has a positive probability to take a new direction that is different from the previous direction. An exception is that a cell always chooses to move in the previous direction whenever it tumbles.. We consider the situation that the probability of turning from v to v is the same as the probability of turning from v to v, i.e., the turning kernel satisfies T (v, v ) = T (v, v). This is true when the environment does not pose one or several primary directions of movement. We note that this assumption is introduced merely to tackle the technical difficulties of the mathematical analysis, and the same general results are expected for situations when this assumption does not hold. 3. Main results When the signal changes slowly compared to the intracellular adaptation, cells have enough time to adapt to the external signal along their trajectories, and thus their internal states stay close to the fully adapted states. Under this condition, the evolution of macroscopic cell density n(x, t) can be well described by the PKS equation (.), and the diffusion rate D n and the chemotactic sensitivity χ can be represented as functions of parameters of the cellbased model. Before presenting the formulas of D n and χ, we first introduce the following notations. Let g(v) be any square integrable function of the velocity v, that is, g(v) L (V ), we define the linear operator A : L (V ) L (V ) in the following way: Z (Ag)(v) = g(v) + T (v, v )g(v ) dv. (3.) V We note that using this notation the right-hand side of (.3) can be written as λ(y)(ap)(x, v, y, t). Here p is treated as a function of v. Based on this observation, A describes the turning behavior of a cell once it decides to tumble. If g(v) is taken to be a constant function, e.g., g(v), then using (.) we obtain Z (Ag)(v) = + T (v, v ) dv = + =. V In fact, the null space of A is the linear subspace of all constant functions, and the range of A is all functions that integrate to, i.e., R V (Ag)(v)dv =. We define the operator B as the pseudo-inverse of A which maps elements in the range of A to its pre-image under A and we choose the pre-image with integral. For any matrix F = (f ij ) i,j q, and any linear operator P : L (V ) L (V ), we denote that F P = (f ij P) i,j q. For example, let I q be the identity matrix of size q, and then we have A I qa = (δ ij A) i,j q = B A A, A

9 Macroscopic equations for bacterial chemotaxis 9 where δ ij = if i = j, and if i j. In addition, let I L (V ) be the identity operator in L (V ), that is, I L (V ) maps any square integrable function to itself. According to this notation, we have ( yf) y=ȳ(s) I L (V ) = ( f i / y j y=ȳ(s) I L (V )) i,j q, where f i is the i-th component of f given in (.). Using the above notations, D n and χ can be represented as D n = Z v Bv dv, (3.) V λ χ(s) = λ V Z V» v ( yλ) y=ȳ(s) λ I qa + ( yf) y=ȳ(s) I L (V ) where denotes tensor product. The invertibility of the operator λ I qa + ( yf) y=ȳ(s) I L (V ) V dȳ(s) ds v dv, (3.3) is guaranteed by the properties of the cell-based model, and this follows from Lemma part (iv) in Section. The negative signs preceding the integrals in (3.) and (3.3) annihilate the negativity of the following integrals so that D n and χ are positively defined. We note that Equation (.) with (3.) and (3.3) are derived under a small signal variation assumption in Section. Albeit this, our simulations in Section 5 show that solutions of the derived equation match the cell-based model quantitatively even this assumption is violated, as long as the external signal changes slowly enough so that most of the cells are close to their adapted state. For signals that change extremely fast, a large fractions of cells can be far from their adapted state, and Equation (.) with (3.) and (3.3) agree with the underlying cell-based model qualitatively. The formulas (3.) and (3.3) give explicit relations of the diffusion coefficient D n and the chemotactic sensitivity χ with properties of the intracellular signaling and cell movement. According to these formulas, D n and χ are in general second-order tensors. The tensor D n given by (3.) is mainly determined by the turning behavior of the cells and the basal tumbling rate. It is independent of the signal unless λ depends on the signal. The chemotactic sensitivity given by (3.3) depends on the signal S nonlinearly, and the nonlinearity enters through local properties of the intercellular dynamics near the adapted state only: the Jacobian ( yf) y=ȳ(s), the variation of the adapted state as a function of the signal dȳ(s) ds, the sensitivity of the flagellar motor ( yλ) y=ȳ(s) and possibly the basal turning rate λ. This is not surprising, because under the small signal variation assumption, cells remain close to the adapted state, and thus the behavior of the internal dynamical system is similar to the linearized system around the adapted state. Although the general formulas (3.) and (3.3) are applicable to extract population behavior from any cell-based models of bacterial chemotaxis that satisfy the criteria specified in Sec 3., they involve calculations of the turning operators A and B which are given in abstract forms. We next consolidate these calculations for the most common situation observed in experiments, where cell movement shows directional persistence [7, ] and the turning kernel only depends on the angle between the run directions before and after a tumble, as considered in [7]. In this case, the tensors (3.) and (3.3) are isotropic, and thus can

10 Chuan Xue be reduced to scalars. For other situations, one can carry out these calculations in a similar way. Since the turning kernel only depends on the angle between the previous direction v and the subsequent direction v, one can represent the turning kernel as T (v, v ) = h(θ), (3.) where θ is the angle between v and v, h(θ) > for all θ [, π]. In this situation, given a fixed velocity v before turning, the average velocity v after turning Z v = T (v, v )v dv V is parallel to v. If we further assume that the cell speed v is constant, i.e., V = s B, where B is the unit sphere centered at the origin with radius, then it is easy to show that the operator A satisfies Av = ( ψ d )v, (3.5) where ψ d = ( v v )/s is the so-called index of directional persistence []. Using (3.5), and noticing that R V vdv =, one obtains Bv = ψ d v. Thus (3.) can be simplified as Z «D n = v vdv. (3.) λ ( ψ d ) V V Again using (3.5), one obtains λ I qa + ( yf) y=ȳ(s) I L (V ) v = λ ( ψ d )I q + ( yf) y=ȳ(s) v, and thus v v. λ I qa + ( yf) y=ȳ(s) I L (V ) = λ ( ψ d )I q ( yf) y=ȳ(s) Therefore (3.3) can be simplified as χ(s) =» ( yλ) y=ȳ(s) λ Finally, for V = s B, one has λ ( ψ d )I q ` yf y=ȳ(s) dȳ(s) ds Z «v vdv. V V (3.7) Z v vdv = s V V N I N, (3.) where N is the space dimension, and I N is the identity matrix of size N. Plugging (3.) into (3.) and (3.7), D n and χ can be further simplified as s D n = Nλ ( ψ d ), χ = s Nλ» ( yλ) y=ȳ(s) λ ( ψ d )I q ` yf y=ȳ(s) dȳ(s) ds. (3.9) In the above formulas, the macroscopic quantities D n and χ are completely determined by the parameters of the cell-based model, which can be determined or inferred in single cell

11 Macroscopic equations for bacterial chemotaxis experiments. Once the parameters for single cells are determined, one can use these formulas to calculate the macroscopic parameters for the population dynamics. The above theory is generally applicable to study any run-and-tumble bacteria. To illustrate the accuracy and usefulness of our theory, we apply it to E. coli chemotaxis in Section 5. Before that, we first present the detailed mathematical analysis that leads to these general results in the next section. For readers that are not interested in the mathematical details, next section can be skipped without affecting the understanding of the rest of the paper. Derivation of the macroscopic equations In this section, we give the detailed derivation of the approximating macroscopic equation (.) and the formulas (3.) and (3.3) for D n and χ summarized in the last section. In Sec., we restate our assumptions on the cell-based model in mathematical terms. In Sec., we introduce a change of variable to the intracellular signaling dynamics and nondimensionalize the cell-based model in order to facilitate our analysis. In Sec.3, we define the small signal variation assumption, and use it to derive a closed moment system. Finally, in Sec., we derive Equation (.) from the closed moment system using asymptotic methods and derive the formulas (3.) and (3.3).. Assumptions on the cell-based model Based on the properties of bacterial chemotaxis, we make the following assumptions of the cell-based model... Assumptions on the internal ODE system We assume that f(y, S) in (.) is twice continuously differentiable in both y and S. This assumption is justified because mathematical models of chemical reaction networks usually consist terms based on mass-action kinetics, Michaelis-Menten functions, or Hill functions which are all differentiable to an arbitrary order in the physiological range of the involved variables, as mentioned in the last section. We also assume that the physiological domain Y is a compact invariant set of the ODE system (.), and this means that the concentrations of the intracellular proteins remain bounded. We define the adapted state of a cell as ȳ = ȳ(s), satisfying f`ȳ, S =. Because y represents concentrations of intracellular proteins that are involved in chemotactic signaling, the adapted state ȳ is nonzero and its components, ȳ i, i q, are non-negative. Due to the robustness and effectiveness of bacterial adaptation, it is reasonable to assume that the system (.) satisfies the following conditions: O: The adapted state ȳ(s) is unique as a function of S [, S max]. O: For any constant S [, S max], all the eigenvalues of the Jacobian matrix F(S ) = ( yf) y=ȳ(s) have negative real parts. Consider the situation that the signal is constant, S = S, Condition O guarantees that the adapted state ȳ(s ) is locally asymptotically stable. Indeed, in mathematical models of bacterial chemotactic signaling, ȳ(s ) is expected to be globally asymptotically stable, which is the second property of the cell-based models specified in Sec 3.

12 Chuan Xue.. Assumptions on the turning rate and turning kernel We assume that the turning rate λ(y) is nonnegative and twice continuously differentiable, and that the basal turning rate λ = λ(ȳ) is strictly positive. In general, λ is allowed to depend on S through ȳ(s). For simplicity of notation, we omit the dependence of λ on S in the remainder of the paper. We define operator T : L (V ) L (V ) as follows: Z (T g)(v) = T (v, v )g(v ) dv, (.) V and denote by K the nonnegative cone of L (V ), i.e., K = {g L (V ) : g }. We assume that the turning kernel T L (V V ) to be any function that have the following properties: T: T (v, v ) = T (v R, v), V T (v, R v ) dv =, V T (v, v ) dv <. T: There are functions u, φ, ψ K with the property u and φ > a.e., such that u (v)φ(v ) T (v, v ) u (v)ψ(v ). T3: T <, where on the left-hand side of the inequality represents a constant function independent of v. These assumptions are all satisfied by the run-and-tumble movements of bacteria in general. From these assumptions, one can prove that: (i) T is a compact, self-adjoint operator on L (V ), with a spectral radius ; (ii) is a simple eigenvalue with normalized eigenfunction g(v) ; and (iii) the eigenfunctions of T form an orthogonal basis of L (V ). The detailed proof can be found in []. Using these notations, the operator A defined in (3.) can be represented as A = I+T. We denote N (A) and R(A) as the null space and the range of A. We follow the notation in Section 3 and denote that F P = (f ij P) i,j q for any matrix F = (f ij ) i,j q, and any linear operator P : L (V ) L (V ). Lemma The operator A has the following properties: (i) A L (V ). (ii) N (A) =, R(A) = = {g L (V ) R V g(v) dv = }. (iii) Nontrivial eigenvalues of A are negative, and the eigenfunctions of A form an orthogonal basis of L (V ). (iv) Let F = (f ij ) i,j q be a constant q q matrix, I q be the identity matrix of size q, and I L (V ) be the identity operator in L (V ). If all eigenvalues of F have negative real part, then F I L (V ) + I qa = (f ij I L (V ) + δ ij A) i,j q is invertible as an operator on `L (V ) q. Proof. (i)-(iii) follows from the properties of T and the definition of A directly. In the following, we prove (iv). Assume that all the eigenvalues of matrix F have negative real parts, then F is invertible and all the eigenvalues of F have negative real parts. Notice that F I L (V ) + I qa = F (I qi L (V ) + F A), thus to prove that (F I L (V ) + I qa) is invertible we only need to show that (I qi L (V ) + F A) is invertible. By (iii), the eigenvalues of A satisfy {µ i : = µ > µ µ 3...}, and the corresponding eigenfunctions {h i (v), i } form an orthogonal basis of L (V ). Therefore

13 Macroscopic equations for bacterial chemotaxis 3 for any g(v) `L (V ) q, we have the decomposition g(v) = X g i h i (v) i= where g i R q. Applying (I qi L (V ) + F A) to g(v), we obtain X X (I qi L (V ) + F A)g(v) = (I qi L (V ) + F A)g i h i (v) = (I q + µ i F )g i h i (v). i= (.) In the last step of the above equation, we used I L (V )h i (v) = h i (v) and Ah i (v) = µ i h i (v). For any g(v) N (I qi L (V ) + F A), using (.) we obtain X (I q + µ i F )g i h i (v) =. i= Because h i (v) form an orthogonal basis in L (V ), we deduce (I q + µ i F )g i = for all i. Furthermore, because all eigenvalues of F have negative real parts and µ i, the matrices (I q + µ i F ) are invertible and thus g i = for all i. Plugging g i = into the decomposition of g(v), we have g(v) = and thus N (I qi L (V ) + F A) is trivial. i=. Change of variables and nondimensionalization of the cell-based model A bacterial cell responds to external signals by measuring the deviation of its internal state y to the adapted state ȳ, which is given by z = y ȳ(s(x, t)). In the absence of signal gradients, the cell remains adapted all the time with z, and its movement is unbiased. In the presence of signal gradients, the variable z is nonzero and depends on time, thus the cell movement is biased. Based on these considerations, we use the linearized variable z to characterize the internal state of a cell in our analysis. Applying a change of variable to (.), one obtains the following system for z, where dz dt = dȳ(s) + dy dt dt = dȳ(s) ds Ṡ + f(z + ȳ, S) (.3) Ṡ = t S + v S is the Lagrangian gradient of the signal along the cell path (v is the cell velocity). Using Taylor s theorem and the fact that f(ȳ, S) =, we can rewrite f(z + ȳ, S) in the following form, f(z + ȳ, S) = F(S)z + R(z, S), (.) with F(S) = ( yf) y=ȳ(s), R(z, S) K z. (.5) Here K is a positive constant proportional to the upper bound of the second derivatives of f(y, S) in y: K max max y iy j f k (y, S) >. S S max y Y i,j,k q

14 Chuan Xue Combining (.3) and (.), and for simplicity of notation denoting we obtain the following equation for z d(s) = dȳ(s) ds, (.) dz = d(s)ṡ + F(S)z + R(z, S), (.7) dt where d(s), F(S), and R(z, S) are given in (.5) and (.). We also rewrite the turning rate λ in terms of z, and expand it as λ(y) = λ(z + ȳ) = λ + b z + r(z), (.) where λ is the basal turning frequency of cells, b = yλ y=ȳ is the sensitivity of the flagella motors to the internal signal, and r(z) is the remaining nonlinear term, satisfying r(z) K z with K max y Y max yi yj λ(ȳ). (.9) i,j q Using the new variable z, we define an equivalent velocity jump process to (.3) p(x, v, z, t) t = h i + x vp(x, v, z, t) h + z Z λ + b z + r(z) p(x, v, z, t) + V d(s)ṡ + F(S)z + R(z, S) i p(x, v, z, t) T (v, v )p(x, v, z, t)dv «. (.) Here we used the same function p to denote the microscopic cell density in terms of the internal variable z for simplicity of notation. In the following, we nondimensionalize Equation (.). To do that, we first nondimensionalize the signal and the internal state variables by setting S = S S, z i = z i Y i, i q, where S is a typical signal scale (e.g, the signal concentration that saturates half of the receptors), and Y i is the maximal range of the variable y i = z i + ȳ i. We then define the nondimensionalized space, velocity, time variables as x = x L a, v = v s, t = t t a, where s is the average cell speed, t a is the adaptation time scale specified below, and L a = s t a is the travel distance of a cell on the adaptation time scale. Bacterial chemotaxis involves multiple time scales. Ligand binding to chemoreceptors on the cell membrane, change of kinase activity, and phosphorylation reactions inside a cell occur within fractions of seconds. Methylation and demethylation of the receptors that cause adaptation of a cell occur on a time scale of seconds. The adaptation time scale is the slowest time scale for intracellular dynamics. It is intrinsically determined by the intracellular signaling network, and reflects the time scale of methylation and demethylation. Let σ m(f(s ))

15 Macroscopic equations for bacterial chemotaxis 5 denote the maximum real part of the eigenvalues of F(S ), where S [, S max] is constant. By Property O of (.), we have σ m(f(s )) negative for all S. We then choose t a to be «t a min σm`f(s ). (.) S S max Since the total cell mass is conserved, we denote it as Z Z Z N = p(x, v, z, t) dxdvdz, and scale the microscopic cell density p by setting p (x, v, z, t ) = p(x, v, z, t)/n. Finally, we nondimensionalize the functions F(S), R(z, S), λ, b and r(z) by setting F (S) = t af(s), R i(z, S) = t ar i (z, S)/Y i, i q, λ = t aλ, b i = b i t ay i, r (z ) = t ar(z), i q. where R i, b i represents the i-th component of the vector R and b. We rewrite (.) using these new nondimensionalized variables, and after dropping the primes for simplicity of notation, the nondimensionalized equation takes the same form as (.). Because cells do not interact mechanically, the variable p(x, v, z, t) can also be thought as the microscopic cell density in the (N + q)-dimensional phase space, and thus we have the relation Z Z n(x, t) = p(x, v, z, t)dzdv. V R q In the remainder of Section, we derive Equation (.) of n(x, t) with (3.) and (3.3) from (.)..3 The small signal variation assumption and derivation of the closed moment system We define the z-moment variables as follows: Z M(x, v, t) = p(x, v, z, t) dz, R Z q M k (x, v, t) = z k p(x, v, z, t) dz, k q, R Z q M ki (x, v, t) = z k z i p(x, v, z, t) dz, k, i q. R q (.) By multiplying (.) by and z k, k q, and then integrating over z, we obtain the following z moment equations: M t + v xm = λ AM + M k t qx b i AM i + R, (.3) i= + v xm k = d k (S)ṠM + qx f ki (S)M i + R k (.) i= +λ AM k + qx b i AM ki + R k, i=

16 Chuan Xue where f ki (S) = yi f yk (ȳ(s)) is the (k, i)-th component of the matrix F(S), b i is the i-th component of the vector b, and Z R = r(z)ap(x, v, z, t) dz, R Z q Z R k = R k (z, S)p(x, v, z, t) dz, R k = z k r(z)ap(x, v, z, t) dz. R q R q (.5) We note that in (.3) and (.), there are no boundary integral terms in z because p(x, v, z, t) is compactly supported as a function of z. This follows from the fact that z represents the deviation of intracellular chemical concentrations from their adapted state, and thus remains bounded all the time. In order to close the system (.3) and (.), we must estimate M ki, R, R k and R k in terms of M and M k. Bacteria can detect an external signal, translate it into an internal signal, and amplify it. When cells have enough time to adapt to the external signal change along their trajectories, their internal states stays close to the fully adapted state. In this case, z remains small for each cell, which means higher-order z-moments are much smaller than lower-order z- moments, and therefore hight-order moments can be dropped to close the system (.3) and (.). One such case is when the signal satisfies a small signal variation assumption, which, roughly speaking, assumes the separation of time scales, t a T s, where t a is given by (.) and T max S S max v V λ b d(s)ṡ A (.) is the time scale for the signal variation interpreted by a cell. Here d(s)ṡ represents the internal signal variation of a cell, and λ b represents the gain of the signal at the flagellar motor level. In the rest of this section, we state the small signal variation assumption rigorously, and show that under this assumption, the magnitudes of the terms involving M ki, R, R k and R k on the right-hand side of (.3) and (.) are relatively small and can be dropped. We start by introducing some notations and definitions. For any small δ >, by Conditions O and O of (.), there exists a tubular neighborhood D δ { z < δ} [, S max] which have the following two properties. First, its cross-section D S δ, defined as DS δ = D δ {S = S }, contains (, S ) as an inner point. Second, on the boundary D S δ, one has F(S )z + R(z, S ) ν <, where ν is the outward normal. We denote ɛ(d δ ) = min S S max min D S δ The choice of D δ is apparently not unique given δ. We define F(S )z + R(z, S ) ν >. (.7) ε δ = max D δ ɛ(d δ ), Dδ = arg max D δ ɛ(d δ ), (.) i.e., Dδ is the δ-tubular neighborhood that gives the largest ɛ, which we define as ε δ. We note that ε δ only depends on δ and the model (.). By (.7) and (.), we have ε δ = ɛ( Dδ ) F(S )z + R(z, S ) ν on DS [, S max]. δ

17 Macroscopic equations for bacterial chemotaxis 7 And since F(S )z + R(z, S ) ν <, this is equivalent to F(S )z + R(z, S ) ν ε δ on DS δ [, S max]. The small signal variation assumption In the following analysis, we assume that the external signal S satisfies max d(s)ṡ ε δ with δ << min{, /K, p λ /K, / b }, (.9) x R 3,v V where ε δ is defined by (.), is the l (R q ) norm, and d(s) is given by (.). Remark For z small enough, the nonlinear terms in (.7) is small, and we have ε δ δ/t a. In this limit, the small signal variation assumption simply states that /T s δ/t a /t a, or equivalently t a T s. Under the assumption (.9), we have, h i d(s)ṡ + F(S)z + R(z, S) ν on DS δ [, S max]. Therefore Dδ is an invariant set of the dynamical system (.7). This means that if all cells start with initial internal state z() Dδ, then they remain to have z(t) Dδ for all t >, regardless of their direction of movement. Or equivalently, if p(x, v, z, ) for z / Dδ, then p(x, v, z, t) for z / Dδ. Lemma Assume that the external signal satisfies the small signal variation assumption (.9) and p(x, v, z, ) for z / Dδ. Then we have M k δm, M ki δ M, R < δ K AM, k, i q, t, R k < δ K M, R k < δ 3 K AM, k q, t. (.) Proof. We only need to show that the estimates (.) hold if p(x, v, z, t) for z / Dδ and for all t. Assume that p(x, v, z, t) for z / Dδ, then the integrations in (.) and (.5) can be restricted to the domain { z < δ}. Using the estimates (.5) and (.9), we immediately obtain the estimates (.). Given the estimates (.), it is easy to see that the magnitude of the terms involving M ki, R,, R k, and R k in (.3) and (.) are much smaller than the leading terms in these equations. For example, by (.9) we have δ K λ, thus by Lemma we have R λ AM. Therefore R is a higher order term in Equation (.3). Based on these observations, we drop the small terms involving M ki, R,, R k, and R k from (.3) and (.) and arrive at the following closed moment system: M t + v xm = λ AM + M k t qx b i AM i, (.) i= + v xm k = d k (S)ṠM + qx f ki (S)M i + λ AM k. (.) The system (.) and (.) represents a mesoscopic model for bacterial chemotaxis. i=

18 Chuan Xue We note that the small signal variation assumption (.9), although easy-to-check analytically, is sufficient but not necessary to derive the estimates (.), and subsequently (.) and (.). A less restrictive requirement is that z remains small for most cells in the signal field, i.e., z < δ for δ << min{, /K, p λ /K, / b }. Given the complexity of the intracellular system (.), interpretation of this looser condition on z in terms of requirements of S(x, t) is rather complicated and model specific. To proceed, we analytically derive Equation (.) and the formulas (3.) and (3.3) for D n and χ under the assumption (.9) in the rest of this section, and numerically investigate to what extent these formulas match the underlying cell-based model when (.9) is violated in Section 5.. The parabolic scaling and derivation of the PKS equation For simplicity of presentation, we assume that the signal S(x) is independent of time in the following, and discuss potential differences in the analysis for general, time-varying signal S(x, t) at the end of this section. Under this assumption, we have Ṡ = S v. For simplicity of notation, we denote ε δ as ε. We introduce the following parabolic scales: t = ε t, x = εx. After rescaling using these scales, the system (.) and (.) can be rewritten in the following compact form: where and ε M t + εv xm = εcm + DM, (.3) M = (M, M,..., M q) T, «λ A b C =, D = T «A. d(s)( S v) λ I qa + F(S)I L (V ) This new space scale is equivalent to the space scale for signal variation, and the quantity d(s)( S v) becomes O(). We next derive the PKS equation as a first order approximation for the cell density n(x, t) using asymptotic analysis. We assume that M can be expanded in powers of ε, i.e., M = M + εm + ε M +, (.) where M i = (M i, M i,..., M i q) T. Here the subscripts are the indices for the moments, and the superscripts indicate the orders of the terms in the expansion. The macroscopic cell density n(x, t) satisfies Z Z Z n(x, t) = p(x, v, z, t) dz dv = M(x, v, t) dv. V R q V Using the expansion (.) for M, we obtain Z n = (M + εm + ε M + ) dv. V

19 Macroscopic equations for bacterial chemotaxis 9 Thus, by expanding n = n + εn + ε n +, we find that Z n i = M i dv, i. V After substituting (.) into the evolution system (.3) and comparing terms with the same order of magnitudes, we have the following systems: O(ε ) : DM =, (.5) O(ε ) : v xm = CM + DM, (.) O(ε ) : M t + v xm = CM + DM, (.7) By Property (iv) of A in Lemma, we have that λ A + F (S) is invertible. Then from (.5) we have AM = and M j =, j q. By Property (ii) of A, we deduce that M is independent of v, i.e., M = M (x, t) and Substituting M = (M,,..., ) T into (.) we have M (x, t) = n (x, t)/ V. (.) v xm «d(s) xs vm = DM λ A b = T «A λ I qa + F(S)I L (V ) Let M be the vector formed by the last N components of M, then the system above can be rewritten as M v xm = λ AM + b A M, (.9) d(s) xs vm = `λ I qa + F(S)I L (V ) M. (.3) Since λ A + F(S) is invertible, we can solve M from (.3) and obtain M = d(s) λ I qa + F(S)I L (V ) xs v M. Substituting this expression into (.9) we obtain, AM = λ v xm λ b A M = v xm d(s) b A λ I qa + F(S)I λ λ L (V ) xs v M. (.3) By Property (ii) of A, zero is a simple eigenvalue with eigenvector. The pseudoinverse operator B of A, as defined in Section 3, can be represented as B = (A ). Applying B to (.3), we obtain, M = B v xm d(s) b λ I qa+f(s)i λ λ L (V ) xs v M +P, (.3) where P, i.e., P = P (x, t), is arbitrary.