A stochastic Keller-Segel model of chemotaxis
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p A stochastic Keller-Segel model of chemotaxis
Pierre-Henri ChavanisNovember 1, 2018
Laboratoire de Physique Th´eorique (CNRS UMR 5152),Universit´e Paul Sabatier,118, route de Narbonne, 31062 Toulouse Cedex 4, FranceE-mail: [email protected]
Abstract
We introduce stochastic models of chemotaxis generalizing the deterministic Keller-Segel model. These models include fluctuations which are important in systems withsmall particle numbers or close to a critical point. Following Dean’s approach, we derivethe exact kinetic equation satisfied by the density distribution of cells. In the mean fieldlimit where statistical correlations between cells are neglected, we recover the Keller-Segelmodel governing the smooth density field. We also consider hydrodynamic and kineticmodels of chemotaxis that take into account the inertia of the particles and lead to adelay in the adjustment of the velocity of cells with the chemotactic gradient. We makethe connection with the Cattaneo model of chemotaxis and the telegraph equation.
In biology, many organisms (bacteria, amoebae, cells,...) or social insects (like ants, swarms,...)interact through the process of chemotaxis [1, 2, 3]. Chemotaxis is a long-range interactionthat accounts for the orientation of individuals along chemical signals that they produce them-selves. Famous examples of biological species experiencing chemotaxis are the slime moldamoebae
Dictyostelium discoideum , the flagellated bacteria
Salmonella typhimurium and
Es-cherichia coli , the human endothelial cells etc. When the interaction is attractive, chemo-taxis is responsible for the self-organization of the system into coherent structures such aspeaks, clusters, aggregates, fruiting bodies, periodic patterns, spirals, rings, spots, honey-comb patterns, stripes or even filaments. This spontaneous organization has been observedin several experiments [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and numerical simulations[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Chemotactic attractionis therefore a leading mechanism to account for the morphogenesis and self-organization ofbiological systems. For example, it has been advocated to explain aggregation patterns inbacteria, tissue organization during embryonic growth, cell guidance, fish skin pigmentationpatterning, angiogenesis in tumour progression and wound healing, formation of plaques inAlzheimer’s disease, dynamics of blood vessel formation etc [24, 34]. It is fascinating to realizethat the self-organization of chemotactic species in biology shares some analogies with the self-organization of galaxies in astrophysics and large-scale vortices (like Jupiter’s great red spot) in1wo-dimensional turbulence . A first successful model of chemotactic aggregation is providedby the Keller-Segel (KS) model [41] introduced in 1970. The standard KS model can be writtenas ∂ρ∂t = ∇ · ( D ∗ ∇ ρ − χρ ∇ c ) , (1) ∂c∂t = D c ∆ c − kc + hρ. (2)It consists in two coupled differential equations that govern the evolution of the density of cells(or other biological entities) ρ ( r , t ) and the evolution of the secreted chemical c ( r , t ). The firstequation (1) is a drift-diffusion equation. The cells diffuse with a diffusion coefficient D ∗ andthey also move in a direction of a gradient of the chemical (chemotactic drift). The chemotacticsensitivity χ is a measure of the strength of the influence of the chemical gradient on the flowof cells. The coefficient χ can be positive or negative. In the first case (chemoattraction), theparticles climb the chemical gradient and form clusters. In the second case (chemorepulsion),they descend the chemical gradient and repell each other. In that case, the chemical acts like apoison. The second equation (2) in the KS model is a reaction-diffusion equation. The chemicalis produced by the bacteria with a rate h and is degraded with a rate k . It also diffuses witha diffusion coefficient D c . When chemotactic attraction prevails over diffusion, the KS modeldescribes a chemotactic collapse leading to aggregates or Dirac peaks. There is a vast literatureon this subject. We refer to Perthame [42] for numerous references in applied mathematics andto Chavanis [43] for additional references in physics.The first equation of the KS model can be interpeted as a mean-field Smoluchowski equationdescribing a system of Brownian particles in interaction. On the other hand, in the limit oflarge diffusivity of the chemical, we can make a quasi-stationary approximation ∂c/∂t ≃ ∂ρ∂t = ∇ · ( D ∗ ∇ ρ − χρ ∇ c ) , (3)∆ c − k c = − λρ, (4)where we have set k = k/D c and λ = h/D c . In the absence of degradation of the chemical( k = 0), the field equation (4) reduces to the Poisson equation ∆ c = − λρ (see [44] andAppendix C of [32] for a precise justification of these approximations). In that case, the Keller-Segel (KS) model becomes isomorphic to the Smoluchowski-Poisson (SP) system ∂ρ∂t = ∇ · (cid:20) ξ (cid:18) k B Tm ∇ ρ + ρ ∇ Φ (cid:19)(cid:21) , (5)∆Φ = S d Gρ, (6) These analogies are intrinsically due to the long-range attractive nature of the interaction. In particular, self-gravitating systems, 2D vortices and chemotactic species interact through a field produced by the distribution ofparticles via a Poisson equation (or its generalizations). Furthermore, the process of self-organization is describedby relatively similar relaxation equations corresponding to nonlinear mean field Fokker-Planck equations [35].Therefore, self-gravitating systems, 2D vortices and chemotactic species share many analogies despite theirvery different physical nature. These striking analogies have been emphasized by the author in several papers[36, 37, 38, 39, 40, 35]. D ∗ = k B T /ξm , χ = 1 /ξ , c = − Φ, λ = S d G . In particular, the concentration of the secreted chemical c ( r , t ) = − Φ( r , t ) in biology plays the role of the gravitational potential (with the opposite sign)in astrophysics . More generally, when we consider a system of Brownian particles interactingvia an arbitrary binary potential u ( r − r ′ ) and make a mean-field approximation [53, 54, 55],we obtain the mean-field Smoluchowski equation ∂ρ∂t = ∇ · (cid:20) ξ (cid:18) k B Tm ∇ ρ + ρ ∇ Φ (cid:19)(cid:21) , (7)Φ( r , t ) = Z ρ ( r ′ , t ) u ( r − r ′ ) d r ′ . (8)The main difference between models (1)-(2) and (7)-(8) comes from the equation for the field c ( r , t ) or Φ( r , t ). Equation (2) is non-markovian since the concentration of the chemical c ( r , t )at time t depends on the concentration of the bacteria and of the chemical at earlier times.By contrast, Eq. (8) is markovian since the potential Φ( r , t ) is assumed to be instantaneouslyproduced by the distribution of particles.It is important to note that the Keller-Segel model is a mean field model which ignoresfluctuations. This implicitly assumes that the number of cells N → + ∞ and that we are farfrom a critical point [56]. Now, in biology, the number of particles in the system can be relativelysmall. Furthermore, from the statistical physics viewpoint, it is natural to investigate the roleof fluctuations during chemotaxis. In order to go beyond the mean field approximation, someauthors [17, 57, 58, 32] have proposed to return to a corpuscular description of the dynamicsand to describe the motion of the particles (chemotactic species or “active” walkers) by N coupled stochastic Langevin equations of the form d r i dt = χ ∇ c d ( r i ( t ) , t ) + p D ∗ R i ( t ) , (9) ∂c d ∂t = D c ∆ c d − kc d + h N X i =1 δ ( r − r i ( t )) , (10)where r i ( t ) denote the positions of the particles, c d ( r , t ) is the exact field of secreted chemicaland R i ( t ) is a white noise satisfying h R i ( t ) i = and h R i,α ( t ) R j,β ( t ′ ) i = δ ij δ αβ δ ( t − t ′ ) where i = 1 , ..., N refer the the particles and α = 1 , ..., d to the dimensions of space. Note that themotion of cells is treated on an individual basis but the chemical signals are treated in thecontinuum limit. This separation of scales appears to be reasonable in most applications. Inthe mean field approximation, these stochastic equations lead to the KS model (1)-(2) . Whenthe reaction-diffusion equation (10) is replaced by a Markovian equation of the form∆ c d − k c d = − λ N X i =1 δ ( r − r i ( t )) , (11) One great achievement of Keller & Segel [41] was to interpret slime mold aggregation as a manifestation ofa fundamental instability in a uniform distribution of amoebae and acrasin (chemoattractant). As noticed in[50, 51], this instability is closely related to the Jeans gravitational instability in astrophysics [52]. Stevens [57] gives the first rigorous derivation (in the mathematical sense) of the KS model from an inter-acting stochastic many-particle system where the interaction between the particles is rescaled in a moderateway as the population size N tends to infinity.
3e obtain a simplified model of chemotaxis that leads to the simplified KS model (3)-(4) inthe mean field approximation. More generally, for Brownian particles interacting via a binarypotential of interaction u ( r − r ′ ), one obtains the stochastic model d r i dt = − ξ ∇ Φ d ( r i ( t ) , t ) + s k B Tξm R i ( t ) , (12)Φ d ( r , t ) = N X i =1 m u ( r − r i ( t )) , (13)considered in [59, 60, 61, 53, 62, 54, 55, 47, 56]. In the mean field approximation [53, 54, 55],these equations yield the mean-field Smoluchowski equation (7)-(8).In systems with weak long-range interactions, the mean field approximation is expectedto become exact in a proper thermodynamic limit N → + ∞ such that the strength of thepotential scales like 1 /N while the volume V remains of order unity [54]. In the context ofchemotaxis, the differences between mean field and non mean field models have been discussedby Grima [34] who showed situations where the mean field approximation fails to predict thewidth of the aggregate sizes. In particular, the disagreement is very severe close to the criticalpoint where we know that mean field approximation breaks down in general [63]. This isbecause the fluctuations become very important so that it is not possible to neglect the two-body correlation function anymore [56]. On the other hand, the mean field approximationassumes that the number of particles N ≫
1. In stellar systems and plasmas, this is always thecase. However, for biological systems, the number of interacting bacteria or cells is frequentlyless than a few thousands so that finite N effects and statistical fluctuations are important.In view of these remarks, it is highly desirable to obtain stochastic kinetic equations that takeinto account fluctuations and that go beyond the deterministic mean field Keller-Segel model.Such equations are discussed in the present paper. In the first part of the paper (Sec. 2),following Dean’s approach [60], we derive the exact kinetic equation satisfied by the densitydistribution of chemotactic species. This equation takes into account stochastic fluctuationsand memory effects present in the field equation for the secreted chemical. If we average overthe noise, we recover the hierarchy of kinetic equations discussed by Newman & Grima [58].If we make a mean-field approximation, we recover the Keller-Segel model [41]. Therefore,this exact stochastic kinetic equation generalizes several models introduced in the chemotacticliterature. We also propose a simplified kinetic equation for a coarse-grained density field(instead of a sum of δ -functions) keeping track of fluctuations. This equation (31)-(32) couldbe of practical interest in chemotaxis. In the second part of the paper (Secs. 3 to 6), we notethat the Keller-Segel model is a parabolic model which neglects the inertia of the particlesand which assumes an instantaneous adjustment of the velocity with the chemotactic gradient.We consider hyperbolic models that generalize this parabolic model. We first consider theCattaneo model of chemotaxis [26] which consists in introducing a delay in the establishmentof the current (Sec. 3). Then, we consider hydrodynamic models including a friction force (Sec.4). Using a semi-linear approximation, we show that the Cattaneo model can be recovered fromthese hydrodynamic equations [56]. In Sec. 5, we generalize these models so as to take intoaccount fluctuations. This leads to stochastic hyperbolic models of chemotaxis which generalizethe ordinary deterministic parabolic Keller-Segel model. Finally, in Sec. 6, we develop a kinetictheory of chemotactic species in phase space taking into account the inertia of the particles andthe discrete nature of the system. We derive stochastic kinetic equations that should improvethe description of the cells’ motion. The link with the parabolic and hyperbolic models is alsodiscussed. 4his paper adapts the results of [56] to the context of chemotaxis with complements andamplification. Although these different stochastic equations (in particular the parabolic ones)are well-known in statistical physics [59, 60, 61, 62, 56] their application to the context ofchemotaxis, proposed in [56], is new and is an important contribution of the present paper. In this section, we introduce a stochastic model of chemotaxis, generalizing the Keller-Segelmodel, by taking into account fluctuations. Let us first derive the exact kinetic equation satisfiedby the density distribution of cells whose dynamics is described by the coupled stochasticLangevin equations (9)-(10). We follow Dean’s approach [60]. The exact density field, expressedin terms of δ -functions, can be written ρ d ( r , t ) = N X i =1 ρ i ( r , t ) = N X i =1 δ ( r − r i ( t )) . (14)For any function F ( r ), we have F ( r i ( t )) = R ρ i ( r , t ) F ( r ) d r . Now, using Ito’s calculus [64], onehas dF ( r i ) dt = Z ρ i ( r , t ) h χ ∇ F ( r ) · ∇ c d ( r , t ) + p D ∗ ∇ F ( r ) · R i ( t ) + D ∗ ∆ F ( r ) i d r . (15)Integrating by parts, we obtain dF ( r i ) dt = Z F ( r ) h − χ ∇ · ( ρ i ( r , t ) ∇ c d ( r , t )) − p D ∗ ∇ · ( ρ i ( r , t ) R i ( t )) + D ∗ ∆ ρ i ( r , t ) i d r . (16)Then, using dF ( r i ) /dt = R ∂ t ρ i ( r , t ) F ( r ) d r and comparing with Eq. (16), we get (using thefact that F is an arbitrary function) ∂ρ i ∂t = − χ ∇ · ( ρ i ( r , t ) ∇ c d ( r , t )) − p D ∗ ∇ · ( ρ i ( r , t ) R i ( t )) + D ∗ ∆ ρ i ( r , t ) . (17)Summing this relation over the i , we finally obtain ∂ρ d ∂t ( r , t ) = D ∗ ∆ ρ d ( r , t ) − χ ∇ · ( ρ d ( r , t ) ∇ c d ( r , t )) − p D ∗ N X i =1 ∇ · ( ρ i ( r , t ) R i ( t )) . (18)Now, the last term can be rewritten [60]: − N X i =1 ∇ · ( ρ i ( r , t ) R i ( t )) = ∇ · ( ρ / d ( r , t ) R ( r , t )) , (19)where R ( r , t ) is a Gaussian random field such that h R ( r , t ) i = and h R α ( r , t ) R β ( r ′ , t ′ ) i = δ αβ δ ( r − r ′ ) δ ( t − t ′ ). Therefore, the system of equations satisfied by the exact density fieldexpressed in terms of δ -functions is ∂ρ d ∂t ( r , t ) = D ∗ ∆ ρ d ( r , t ) − χ ∇ · ( ρ d ( r , t ) ∇ c d ( r , t )) + ∇ · (cid:16)p D ∗ ρ d ( r , t ) R ( r , t ) (cid:17) , (20) ∂c d ∂t = D c ∆ c d ( r , t ) − kc d ( r , t ) + hρ d ( r , t ) . (21)5he first and third terms in the r.h.s. of Eq. (20) correspond to a pure Brownian motionand the second term takes into account chemotaxis, i.e the attraction or repulsion of the cellsby the chemical. As noted by Dean [60], the noise in Eq. (20) appears not additively butmultiplicatively.Integrating Eq. (10), the concentration of the chemical can be expressed in terms of thecell paths as [58]: c d ( r , t ) = h Z d r ′ Z t dt ′ G ( r − r ′ , t − t ′ ) N X i =1 δ ( r ′ − r i ( t ′ )) , (22)where the Green function for the chemical diffusion equation is given by G ( r , t ) = (4 πD c t ) − d/ exp (cid:20) − r D c t − kt (cid:21) . (23)The gradient of the concentration field is ∇ c d ( r , t ) = h Z d r ′ Z t dt ′ ∇ G ( r − r ′ , t − t ′ ) ρ d ( r ′ , t ′ ) . (24)Substituting Eq. (24) in Eq. (20), we obtain ∂ρ d ∂t ( r , t ) = D ∗ ∆ ρ d ( r , t ) − χh ∇ · (cid:20) ρ d ( r , t ) ∇ Z d r ′ Z t dt ′ G ( r − r ′ , t − t ′ ) ρ d ( r ′ , t ′ ) (cid:21) + ∇ · (cid:16)p D ∗ ρ d ( r , t ) R ( r , t ) (cid:17) . (25)If we average over the noise and introduce the smooth density ρ ( r , t ) = h ρ d ( r , t ) i , we recoverEq. (9) of Newman & Grima [58]: ∂ρ∂t ( r , t ) = D ∗ ∆ ρ ( r , t ) − χh ∇ · Z d r ′ Z t dt ′ [ ∇ G ( r − r ′ , t − t ′ )] h ρ d ( r , t ) ρ d ( r ′ , t ′ ) i . (26)If we make a mean field approximation h ρ d ( r , t ) ρ d ( r ′ , t ′ ) i ≃ ρ ( r , t ) ρ ( r ′ , t ′ ) in Eq. (26), we recoverthe Keller-Segel model [41]: ∂ρ∂t ( r , t ) = D ∗ ∆ ρ ( r , t ) − χ ∇ · ( ρ ( r , t ) ∇ c ( r , t )) , (27)with c ( r , t ) = h Z d r ′ Z t dt ′ G ( r − r ′ , t − t ′ ) ρ ( r ′ , t ′ ) . (28)Given the definition of the Green function G , the smooth concentration c ( r , t ) is solution of thereaction-diffusion equation ∂c∂t = D c ∆ c − kc + hρ. (29)We also note, for future reference, that the steady solutions of the KS model (27) correspondto a mean field Boltzmann-like distribution ρ = Ae c/T eff , (30)6here T eff = D ∗ /χ is an effective temperature given by an Einstein relation.Grima [34] has shown that the mean field approximation may lead to wrong results if weare close to a critical point or if the number of particles is not large enough. Therefore, itmay be useful to have a more general model than the Keller-Segel model (1)-(2) which keepstrack of fluctuations. Equations (20)-(21) are exact and contain the same information as the N -body stochastic Langevin equations (9)-(10). They are not very useful for practical purposessince they govern the evolution of a density field which is expressed as a sum of δ -functions.It is easier to directly solve the equivalent N -body stochastic Langevin equations (9)-(10).However, using phenomenological arguments like those described in [62, 56], we can consider aspatio-temporal coarse-grained distribution ρ ( r , t ) which smoothes out the exact density field ρ d ( r , t ) while keeping track of fluctuations. We also assume that the spatio-temporal windowis sufficiently small so that we can make the approximation ρ (2) ( r , r ′ , t ) ≃ ρ ( r , t ) ρ ( r ′ , t ). In thatcase, we obtain the stochastic Keller-Segel model for the coarse-grained distribution ∂ρ∂t ( r , t ) = D ∗ ∆ ρ ( r , t ) − χ ∇ · ( ρ ( r , t ) ∇ c ( r , t )) + ∇ · (cid:16)p D ∗ ρ ( r , t ) R ( r , t ) (cid:17) , (31) ∂c∂t = D c ∆ c − kc + hρ, (32)generalizing the deterministic Keller-Segel model (1)-(2). This equation is one of the mostimportant result of this paper. As shown in Appendix B of [56], the form of the noise termin Eq. (31) can be obtained from the general theory of fluctuations developed in Landau &Lifshitz [65]. This provides another, direct, justification of the stochastic Eq. (31). As shownin [56], the mean field approximation breaks down close to a critical point because the two-body correlation function diverges. In that case, it may be more relevant to use the stochasticKeller-Segel model (31)-(32) including fluctuations instead of the deterministic Keller-Segelmodel (1)-(2).It is also very important to take into account fluctuations when the system can be found inseveral metastable states. If we introduce the coarse-grained free energy functional F c.g. [ ρ, c ] = D ∗ χ Z ρ ln ρ d r + 12 h Z (cid:2) D c ( ∇ c ) + kc (cid:3) d r − Z ρ c d r , (33)we can write the stochastic equation (31) in the form ∂ρ∂t = ∇ · (cid:20) χρ ( r , t ) ∇ δF c.g. δρ (cid:21) + ∇ · (cid:16)p D ∗ ρ ( r , t ) R ( r , t ) (cid:17) . (34)This equation can be viewed as a Langevin equation for the field ρ ( r , t ). The evolution of theprobability of the density distribution W [ ρ, t ] is governed by a Fokker-Planck equation of theform ∂W [ ρ, t ] ∂t = − Z δδρ ( r , t ) (cid:26) ∇ · ρ ( r , t ) ∇ (cid:20) D ∗ δδρ + χ δF c.g. δρ (cid:21) W [ ρ, t ] (cid:27) d r . (35)At equilibrium, we have W [ ρ ] ∝ e − F c.g. [ ρ ] /T eff − α R ρd r with F c.g. [ ρ ] = D ∗ χ R ρ ln ρ d r − R ρ c d r (we have substituted Eq. (32) with ∂c/∂t = 0 in Eq. (33)). For N → + ∞ , the equilibriumdistribution W [ ρ ] is strongly peaked around the global minimum of F c.g. [ ρ ] at fixed mass M = R ρ d r . However, the system can remain trapped in a metastable state (local minimum of7 c.g. [ ρ ]) for a very long time which becomes infinite at the thermodynamic limit N → + ∞ . Letus be more precise. If we ignore the noise term, Eq. (34) reduces to ∂ρ∂t = ∇ · (cid:20) χρ ( r , t ) ∇ δF c.g. δρ (cid:21) , (36)which is the deterministic Keller-Segel model (27). This equation satisfies an H-theorem˙ F = − Z χρ ( D ∗ ∇ ρ − χρ ∇ c ) d r − h Z ( D c ∆ c − kc + hρ ) d r ≤ , (37)with ˙ F = 0 iff the distribution is given by Eq. (30). Therefore, a steady state is stable iff it is a(local) minimum of free energy at fixed mass. Assuming that the free energy is bounded frombelow, we know from Lyapunov’s direct method that the system will relax towards a steadystate that is a minimum (global or local) of the free energy functional F c.g. [ ρ ] at fixed mass(maxima or saddle points of free energy are linearly dynamically unstable with respect to meanfield Fokker-Planck equations [35]). If the free energy admits several local minima, the selectionof the steady state will depend on a notion of basin of attraction . Without noise, the systemremains on a minimum of free energy forever. Now, in the presence of noise, the fluctuationscan induce dynamical phase transitions from one minimum to the other. We should thereforesee the system “jump” between different states. Thus, accounting correctly for fluctuationsis very important when there exists metastable states. The probability of transition scalesas e − ∆ F/T eff where ∆ F is the barrier of free energy between two minima. Therefore, on aninfinite time, the system will explore all the minima and will spend most time in the globalminimum. This will be the case only if N is not too large. Indeed, for systems with long-rangeinteractions, the barrier of free energy ∆ F scales like N so that the probability of escape froma local minimum is very small and behaves like e − N . Therefore, even if the global minimum isin principle the most probable state, metastable states are highly robust in practice since theirlifetime scales like e N . They are thus fully relevant for N ≫
1: metastable states are in practice“stable states”. These interesting features (basin of attraction, dynamical phase transitions,metastability,...) would be interesting to study in more detail in the case of chemotaxis. Thestudy of the stochastic Keller-Segel model will be considered in future publications.
The general Keller-Segel (GKS) model [41] can be written as ξ ∂ρ∂t = ∇ · ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) , (38) ∂c∂t = D c ∆ c − k ( c ) c + h ( c ) ρ, (39)where D = D ( ρ, c ) and D = D ( ρ, c ) can both depend on the concentration of the cellsand of the chemical. This takes into account microscopic constraints, like close-packing effects,that can hinder the movement of cells and lead to nonlinear diffusion and nonlinear mobility.The GKS model (38)-(39) can be viewed as a nonlinear mean field Fokker-Planck equation associated with a notion of effective generalized thermodynamics [35]. The first equation canbe written in the form of a continuity equation ∂ t ρ = −∇ · J with a current J = − ξ ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) . (40)8t is important to note that the GKS model is a parabolic model like the usual heat diffusionequation. Like for the Fourier law of heat conduction, it is assumed that the current J isinstantaneously equal to the right hand side of Eq. (40), that we shall call the “chemotacticgradient” for future reference. In the context of heat conduction, Cattaneo [66] has proposeda modification of Fourier’s law in order to describe heat propagation with finite speed. In thecontext of chemotaxis, Dolak & Hillen [26] have introduced a Cattaneo model for chemosensitivemovement. They assume that the current is not instantaneously equal to the chemotacticgradient but relaxes to it with a time constant 1 /τ . Then, the corresponding Cattaneo modelfor chemosensitive movement reads ∂ρ∂t + ∇ · J = 0 , (41) τ ∂ J ∂t + J = − ξ ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) . (42)Taking the time derivative of Eq. (41) and using Eq. (42), we obtain the hyperbolic model τ ∂ ρ∂t + ∂ρ∂t = 1 ξ ∇ · ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) . (43)This equation, which is second order in time, is analogous to the telegraph equation whichgeneralizes the diffusion equation by introducing memory effects. For τ = 0, we recover theGKS model (38)-(39) as a particular case. The parabolic Keller-Segel model [41] is able to reproduce the formation of clusters (clumps)resulting from chemotactic collapse. This can explain experiments on bacteria like
Escherichiacoli or amoebae like
Dictyostelium discoideum exhibiting pointwise concentrations [4, 14, 9, 11,5, 6, 7, 8]. Recently, several experiments with human endothelial cells have shown the formationof networks that can be interpreted as the initiation of a vasculature [10, 13, 27, 16, 15]. Cellsrandomly spread on a gel matrix autonomously organize to form a continuous multicellularnetwork which can be described as a collection of nodes connected by chords [27]. This processtakes place during the early stages of vasculogenesis in embryo development. These filamentsare observed in the experiments of capillary blood vessel formation. These structures cannot beexplained by the Keller-Segel parabolic model which generically leads to pointwise blow-up .In order to account for these filaments, hyperbolic models of chemotaxis have been introduced[27, 67, 38, 29, 50, 32, 68]. They have the form of damped hydrodynamic equations taking In fact, some Keller-Segel models including cell kinetics can, under certain conditions, give rise to network-like patterns (see Fig. 12c of [24]). The type of hydrodynamic equations (44)-(45) including a long-range mean field interaction, a densitydependent pressure and a friction force were introduced in Chavanis [69] (see also [36]) for Langevin particles ininteraction and called the damped Euler equations . Their application to chemotaxis and gravity was mentioned.These equations can be derived from kinetic equations (nonlinear mean-field Fokker-Planck equations) by using alocal thermodynamic equilibrium condition (L.T.E.) to close the hierarchy of hydrodynamic moments [69, 47, 32].However, they remain heuristic because the L.T.E. approximation is not rigorously justified. By contrast, inthe strong friction limit ξ → + ∞ , we can rigorously derive the GKS model (38)-(39), also called the generalizedSmoluchowski equation , by using a Chapman-Enskog expansion [70] or a method of moments [71, 32, 35]. Themodel considered by Gamba et al. [27] (see also [67, 15, 29]) corresponds to ξ = 0 in Eq. (45). It can be derivedin an asymptotic limit of kinetic equations of a different type (see [29] and Appendix D of [32]). In more recentpapers [68], the aforementioned authors have also included a friction force in their model. ∂ρ∂t + ∇ · ( ρ u ) = 0 , (44) ∂∂t ( ρ u ) + ∇ ( ρ u ⊗ u ) = − D ( ρ, c ) ∇ ρ + D ( ρ, c ) ∇ c − ξρ u . (45)Considering the momentum equation (45), the inertial term (l.h.s.) models cells directionalpersistence, i.e. the natural tendency of a particle to continue in a given direction in the absenceof any interaction. When D ( ρ, c ) depends only on the density, the first term on the r.h.s. canbe interpreted as a barotropic pressure force −∇ p ( ρ ) (see [35] for different examples of equationsof state). The pressure law is expected to be linear for low densities and to increase rapidlyabove a certain threshold ∼ σ in order to describe the fact that the cells do not interpenetrate.For example, in [69, 40, 32] we proposed to take p ( ρ ) = − σ T eff ln(1 − ρ/σ ) which returnsthe “isothermal” equation of state p = ρT eff for dilute systems ρ ≪ σ where the motion ofan individual cell is not impeded by the other cells [25], and which diverges when the cells arecompressed towards the maximum density ρ → σ . Another possible equation of state is thepolytropic one p ( ρ ) = Kρ γ [72, 32] taking into account anomalous transport (normal transportcorresponds to the isothermal case γ = 1). The chemotactic response D ( ρ, c ) of the bacteriumto the chemical gradient (second term in the r.h.s. of Eq. (45)) can also depend on c and ρ so asto take into account anomalous reactivity (the normal case corresponds to D ( ρ, c ) = ρ but theform D ( ρ, c ) = ρ (1 − ρ/σ ) has also been considered to take into account volume filling effects[24, 69, 40, 35]). Finally, the last term in the r.h.s. of Eq. (45) is a friction force that measuresthe importance of inertial effects. It parametrizes the tendency of the organisms to continuein a given direction. In this inertial model, the velocity of a particle takes a finite time ξ − toget aligned with the chemotactic gradient while in the Keller-Segel model, this alignement isassumed to be instantaneous (see below). The “delay” in the alignement of the velocity withthe chemotactic gradient is similar to the idea that is at the heart of the Cattaneo model inSec. 3.If we neglect the friction force ( ξ = 0) we recover the model introduced by Gamba et al. [27].Alternatively, if we neglect the inertial term (l.h.s.) in Eq. (45) and substitute the resultingexpression [69, 50, 32]: ρ u = − ξ ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) , (46)in Eq. (44), we recover the GKS model. This is valid in a strong friction limit ξ → + ∞ . Wecan also obtain a more general model taking into account some memory effects. If we neglectonly the nonlinear term ∇ ( ρ u ⊗ u ) in Eq. (45), we obtain ∂∂t ( ρ u ) = − D ( ρ, c ) ∇ ρ + D ( ρ, c ) ∇ c − ξρ u , (47)which is equivalent to the Cattaneo model (42) with τ = 1 /ξ . Taking the time derivative of Eq.(44) and substituting Eq. (47) in the resulting expression, we obtain a simplified hyperbolicmodel keeping track of memory effects ∂ ρ∂t + ξ ∂ρ∂t = ∇ · ( D ( ρ, c ) ∇ ρ − D ( ρ, c ) ∇ c ) . (48)This provides a new justification (see also [56]) of the Cattaneo model of chemotaxis from thedamped hydrodynamics equation (44)-(45). This can be viewed as a semi-linear hydrodynamicmodel since its derivation assumes that the nonlinear term ∇ ( ρ u ⊗ u ) in Eq. (45) can beneglected while the full nonlinearities in the r.h.s. of Eq. (45) are taken into account.10 Stochastic hydrodynamic models of chemotaxis
In this section, we generalize the previous hydrodynamic equations in order to take into accountfluctuations. We restrict ourselves to the standard situation where D = ξD ∗ and D = ρ . Thestochastic damped Euler equations generalizing Eqs. (44)-(45) can be written ∂ρ∂t + ∇ · ( ρ u ) = 0 , (49) ∂∂t ( ρ u ) + ∇ ( ρ u ⊗ u ) = − ξD ∗ ∇ ρ + ρ ∇ c − ξρ u − p D ∗ ξ ρ R ( r , t ) . (50)As shown in Appendix B of [56], the form of the noise in these equations can be obtained byapplying the general theory of fluctuations developed by Landau & Lifshitz [65]. If we neglectthe inertial term (l.h.s.) in Eq. (50) and substitute the resulting expression ρ u = − ( D ∗ ∇ ρ − χρ ∇ c ) − p D ∗ ρ R ( r , t ) , (51)where χ = 1 /ξ in Eq. (49), we recover the stochastic Keller-Segel equation (31). This is validin a strong friction limit ξ → + ∞ with ξD ∗ ∼
1. As in Sec. 4, we can obtain a more generalmodel taking into account some memory effects. Indeed, if we neglect only the nonlinear term ∇ ( ρ u ⊗ u ) in Eq. (50), we find χ ∂∂t ( ρ u ) = − D ∗ ∇ ρ + χρ ∇ c − ρ u − p D ∗ ρ R ( r , t ) . (52)Taking the time derivative of Eq. (49) and substituting Eq. (52) in the resulting expression,we obtain the stochastic Cattaneo model of chemotaxis χ ∂ ρ∂t + ∂ρ∂t = ∇ · ( D ∗ ∇ ρ − χρ ∇ c ) + ∇ · ( p D ∗ ρ R ) . (53) In order to take into account fluctuations in a rigorous way, we must start from a microscopicdescription of the dynamics of the chemotactic species. In Sec. 2, we have considered anoverdamped dynamics. However, according to recent observations in biology (as discussed inSec. 4), it is important to take into account the inertia of the particles. A kinetic modelof chemotaxis taking into account finite N effects and inertial effects has been proposed inChavanis & Sire [32]. In the simplest case, the motion of the biological entities is described by N coupled stochastic Langevin equations of the form d r i dt = v i , (54) d v i dt = − ξ v i + ∇ c d ( r i ( t ) , t ) + √ D R i ( t ) , (55) ∂c d ∂t = D c ∆ c d − kc d + h N X i =1 δ ( r − r i ( t )) , (56)11here ξ is a friction coefficient and D a diffusion coefficient in velocity space. We can intro-duce an effective temperature T eff through the Einstein relation T eff = D/ξ [32, 35]. Theoverdamped stochastic equations (9)-(10) can be recovered in a strong friction limit ξ → + ∞ ,neglecting the inertial term in Eq. (55), and writing χ = 1 /ξ and D ∗ = D/ξ . We now pro-ceed in deriving the exact kinetic equation satisfied by the distribution function of cells whosedynamics is described by the coupled stochastic Langevin equations (54)-(56). The exact dis-tribution function, expressed in terms of δ -functions, can be written f d ( r , v , t ) = N X i =1 f i ( r , v , t ) = N X i =1 δ ( r − r i ( t )) δ ( v − v i ( t )) . (57)For any function F ( r , v ), we have F ( r i ( t ) , v i ( t )) = R f i ( r , v , t ) F ( r , v ) d r d v . Now, using Ito’scalculus, one has dF ( r i , v i ) dt = Z f i ( r , v , t ) (cid:20) ∇ r F ( r , v ) · v − ξ ∇ v F ( r , v ) · v + ∇ v F ( r , v ) · ∇ c d ( r , t )+ √ D ∇ v F ( r , v ) · R i ( t ) + D ∆ v F ( r , v ) (cid:21) d r d v . (58)Integrating by parts, we obtain dF ( r i , v i ) dt = Z F ( r , v ) (cid:20) − v · ∂f i ∂ r ( r , v , t ) + ξ ∂∂ v · ( f i ( r , v , t ) v ) − ∇ c d ( r , t ) · ∂f i ∂ v ( r , v , t ) −√ D ∂∂ v ( f i ( r , v , t ) R i ( t )) + D ∆ v f i ( r , v , t ) (cid:21) d r d v . (59)Then, using dF ( r i , v i ) /dt = R ∂ t f i ( r , v , t ) F ( r , v ) d r d v and comparing with Eq. (59), we get ∂f i ∂t + v · ∂f i ∂ r + ∇ c d · ∂f i ∂ v = ∂∂ v · (cid:18) D ∂f i ∂ v + ξf i v (cid:19) − √ D ∂∂ v · ( f i R i ) . (60)Summing this relation over the i , we finally obtain ∂f d ∂t + v · ∂f d ∂ r + ∇ c d · ∂f d ∂ v = ∂∂ v · (cid:18) D ∂f d ∂ v + ξf d v (cid:19) − √ D N X i =1 ∂∂ v · ( f i R i ) . (61)Now, proceeding like in [60], the last term can be rewritten: − N X i =1 ∂∂ v · ( f i ( r , v , t ) R i ( t )) = ∂∂ v · ( f / d ( r , v , t ) Q ( r , v , t )) , (62)where Q ( r , v , t ) is a Gaussian random field such that h Q ( r , v , t ) i = and h Q α ( r , v , t ) Q β ( r ′ , v ′ , t ′ ) i = δ αβ δ ( r − r ′ ) δ ( v − v ′ ) δ ( t − t ′ ). Therefore, the system of equations satisfied by the exact dis-tribution function expressed in terms of δ -functions is ∂f d ∂t + v · ∂f d ∂ r + ∇ c d · ∂f d ∂ v = ∂∂ v · (cid:18) D ∂f d ∂ v + ξf d v (cid:19) + ∂∂ v · (cid:16)p Df d Q ( r , v , t ) (cid:17) , (63) ∂c d ∂t = D c ∆ c d − kc d + h Z f d ( r , v , t ) d v . (64)12his will be called the stochastic Kramers equation of chemotaxis for the exact distributionfunction. Using Eq. (24), it can be written ∂f d ∂t + v · ∂f d ∂ r + h Z d r ′ d v ′ Z t dt ′ ∇ G ( r − r ′ , t − t ′ ) f d ( r ′ , v ′ , t ′ ) · ∂f d ∂ v ( r , v , t )= ∂∂ v · (cid:18) D ∂f∂ v + ξf v (cid:19) + ∂∂ v · (cid:16)p Df d Q ( r , v , t ) (cid:17) . (65)If we average over the noise and introduce the smooth distribution function f ( r , v , t ) = h f d ( r , v , t ) i ,we recover Eq. (60) of Chavanis & Sire [32]: ∂f∂t + v · ∂f∂ r + h ∂∂ v · Z d r ′ d v ′ Z t dt ′ ∇ G ( r − r ′ , t − t ′ ) h f d ( r , v , t ) f d ( r ′ , v ′ , t ′ ) i = ∂∂ v · (cid:18) D ∂f∂ v + ξf v (cid:19) . (66)If we make a mean field approximation h f d ( r , v , t ) f d ( r ′ , v ′ , t ′ ) i ≃ f ( r , v , t ) f ( r ′ , v ′ , t ′ ), we recoverEqs. (66)-(68) of Chavanis & Sire [32]: ∂f∂t + v · ∂f∂ r + ∇ c · ∂f∂ v = ∂∂ v · (cid:18) D ∂f∂ v + ξf v (cid:19) , (67) ∂c∂t = D c ∆ c − kc + h Z f ( r , v , t ) d v . (68)This can be viewed as a mean field Kramers equation of chemotaxis in the same way that theKeller-Segel model can be viewed as a Smoluchowski equation of chemotaxis. In fact, the Keller-Segel model (1)-(2) can be recovered from Eqs. (67)-(68) in a strong friction limit ξ → + ∞ by using a Chapman-Enskog expansion [70] or a method of moments [32]. Let us note, forfuture reference, that the steady solutions of the mean field Kramers equation of chemotaxiscorrespond to a mean field Maxwell-Boltzmann-like distribution f = A ′ e − β ( v / − c ) , (69)where β = 1 /T eff is the inverse effective temperature. If we integrate this distribution overthe velocitities we recover the distribution (30) that is the steady solution of the Keller-Segelmodel (1)-(2).As discussed in the Introduction, the mean field approximation may not always give agood description of the dynamics. On the other hand, Eqs. (63)-(64) for the distributionfunction expressed in terms of δ -functions are exact but they are too complicated for practicalpurposes because they contain exactly the same information as the N -body stochastic Langevinequations (54)-(56). Therefore, as in Sec. 2, we shall introduce a simplified kinetic equationfor a coarse-grained distribution function f ( r , v , t ) which smoothes out the exact distributionfunction f d ( r , v , t ) while keeping track of fluctuations. We propose the simplified stochasticmodel ∂f∂t + v · ∂f∂ r + ∇ c · ∂f∂ v = ∂∂ v · (cid:18) D ∂f∂ v + ξf v (cid:19) + ∂∂ v · (cid:18)q Df Q ( r , v , t ) (cid:19) , (70) ∂c∂t = D c ∆ c − kc + h Z f ( r , v , t ) d v . (71)13his model takes into account inertial effects and fluctuations so that it should provide a gooddescription of the dynamics of chemotactic species. As shown in Appendix B of [56], the formof the noise in these equations can be obtained by applying the general theory of fluctuationsdeveloped by Landau & Lifshitz [65].Let us try to make a connexion with the hydrodynamic equations introduced phenomeno-logically in Sec. 5. Taking the hydrodynamic moments of the stochastic Kramers equation (70)and proceeding as in [32], we obtain ∂ρ∂t + ∇ · ( ρ u ) = 0 , (72) ∂∂t ( ρu i ) + ∂∂x j ( ρu i u j ) = − ∂P ij ∂x j + ρ ∂c∂x i − ξρu i − Z p Df Q i d v , (73)where ρ ( r , t ) = R f d v is the density, u ( r , t ) = (1 /ρ ) R f v d v is the local velocity, w = v − u ( r , t ) is the relative velocity and P ij = R f w i w j d v is the pressure tensor. Defining g ( r , t ) ≡ R √ Df Q d v , it is clear that g is a Gaussian noise and that its correlation function is h g i ( r , t ) g j ( r ′ , t ′ ) i = 2 D Z p f ( r , v , t ) f ( r ′ , v ′ , t ′ ) h Q i ( r , v , t ) Q j ( r ′ , v ′ , t ′ ) i d v d v ′ = 2 Dδ ij δ ( r − r ′ ) δ ( t − t ′ ) Z f ( r , v , t ) d v = 2 Dδ ij δ ( r − r ′ ) δ ( t − t ′ ) ρ ( r , t ) . (74)Therefore, the equation for the momentum (73) can be rewritten ∂∂t ( ρu i ) + ∂∂x j ( ρu i u j ) = − ∂P ij ∂x j + ρ ∂c∂x i − ξρu i − p DρR i ( r , t ) . (75)This equation is not closed since the pressure tensor depends on the next order moment ofthe velocity. If, following [32], we make a local thermodynamic equilibrium (L.T.E.) approx-imation f LT E ( r , v , t ) ≃ ( β/ π ) d/ ρ ( r , t ) e − βw / to compute the pressure tensor, we find that P ij ≃ T eff ρδ ij . In that case, Eqs. (72) and (75) return the stochastic damped Euler equa-tions (49)-(50). We recall, however, that there is no rigorous justification for this local ther-modynamic equilibrium approximation. Therefore, it does not appear possible to rigorouslyderive the damped hydrodynamic equations (49)-(50) from the Kramers equation (70)-(71)by a systematic procedure. Alternatively, if we consider the strong friction limit ξ → + ∞ for fixed β , implying D = ξ/β → + ∞ , the first term in the r.h.s. of Eq. (70) implies that f ( r , v , t ) ≃ ( β/ π ) d/ ρ ( r , t ) e − βv / + O (1 /ξ ), u = O (1 /ξ ) and P ij = T eff ρδ ij + O (1 /ξ ) [32]. Toleading order in 1 /ξ , Eq. (75) becomes ρ u ≃ − ξ (cid:16) T eff ∇ ρ − ρ ∇ c + p Dρ R ( r , t ) (cid:17) . (76)Inserting Eq. (76) in the continuity equation (72) and recalling that T eff = D/ξ = ξD ∗ and χ = 1 /ξ , we recover the stochastic Keller-Segel model (31)-(32). It is therefore possibleto rigorously derive the stochastic Keller-Segel model (31)-(32) from the stochastic Kramersequation (70)-(71) in the strong friction limit ξ → + ∞ . In this paper, we have derived generalized Keller-Segel models of chemotaxis taking into ac-count fluctuations. This leads to stochastic kinetic equations instead of deterministic equations.14luctuations become important close to a critical point [63, 34, 56], so it is valuable to havea model of chemotaxis going beyond the mean field approximation and taking into accountfluctuations. The divergence of the spatial correlation function close to the critical point hasbeen analyzed in detail in [56] for Brownian particles interacting through a binary potential.These particles are described by a stochastic Smoluchowski equation coupled to the markovianfield equation (8). The general methods developed in [56] can be extended to the stochasticKeller-Segel model (31) coupled to the non-Markovian field equation (2). Accounting for fluctu-ations is also important when the number of particles N is small and when there exists severalmetastable states. In that case, fluctuations can trigger dynamical phase transitions from onestate to the other.We have also introduced kinetic models of chemotaxis in phase space taking into accountinertial effects. In the strong friction limit, we recover the Keller-Segel model describing an over-damped dynamics. We have discussed the relation between the kinetic equations in phase spaceand the hydrodynamic equations introduced phenomenologically. Finally, we have shown howthe Cattaneo model of chemotaxis [26] could be obtained from these hydrodynamic equations.This paper and [56] are the first attempts to include fluctuations in the kinetic equations ofchemotaxis (the main results were given in [56] and they have been discussed here specificallywith more details and amplification). In view of the importance of the Keller-Segel model inbiology, the stochastic equations that we propose can have a lot of applications and can openthe way to many new investigations. Their detailed numerical and analytical study is thereforeof considerable interest. We hope to come to these problems in future works. Note added:
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