Flow-driven instabilities during pattern formation of Dictyostelium discoideum

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1 PAPER OPEN ACCESS Flow-driven instabilities during pattern formation of Dictyostelium discoideum To cite this article: A Gholami et al New J. Phys. 0 Manuscript version: Accepted Manuscript Accepted Manuscript is the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an Accepted Manuscript watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors This Accepted Manuscript is IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. As the Version of Record of this article is going to be / has been published on a gold open access basis under a CC BY.0 licence, this Accepted Manuscript is available for reuse under a CC BY.0 licence immediately. Everyone is permitted to use all or part of the original content in this article, provided that they adhere to all the terms of the licence Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions may be required. All third party content is fully copyright protected and is not published on a gold open access basis under a CC BY licence, unless that is specifically stated in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address... on 0/0/ at 0:

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3 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum. Introduction A. Gholami, O. Steinbock, V. Zykov, and E. Bodenschatz,, Max Planck Institute for Dynamics and Self-Organization, Am Fassberg, D-0 Göttingen, Germany Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida -, USA Institute for Nonlinear Dynamics, University of Göttingen, D-0 Göttingen, Germany Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA Abstract. The slime mold Dictyostelium discoideum is a well known model system for the study of biological pattern formation. In the natural environment, aggregating populations of starving Dictyostelium discoideum cells may experience fluid flows that can profoundly change the underlying wave generation process. Here we study the effect of advection on the pattern formation in a colony of homogeneously distributed Dictyostelium discoideum cells described by the standard Martiel-Goldbeter model. The external flow advects the signaling molecule camp downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. The evolution of small perturbations in camp concentrations is studied analytically in the linear regime and by corresponding numerical simulations. We show that flow can significantly influence the dynamics of the system and lead to a flowdriven instability that initiate downstream traveling camp waves. We also show that boundary conditions have a significant effect on the observed patterns and can lead to a new kind of instability. In reaction-diffusion systems, an advective flow can induce unique emergent phenomena. One well known example is the differential flow induced chemical instability (DIFICI) that destabilizes an otherwise spatially homogeneous state of a system. DIFICI was originally predicted by Rovinsky and Menzinger[, ] and later confirmed experimentally for the Belousov-Zhabotinsky (BZ) reaction [, ]. The basic idea behind DIFICI is that the reacting species flow at different speeds. This differential transport can initiate instabilities of the homogeneous steady state that induce propagating wave trains moving in the flow direction. Later, conditions for the emergence of three-dimensional spatial and spatiotemporal patterns due to DIFICI were derived by Malchow [, ] and demonstrated by patterns in a minimal phytoplankton-zooplankton model []. Instabilities in the homogeneous distribution can arise if phytoplankton and zooplankton

4 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R Page of 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum move with different velocities, regardless of which one is faster. This mechanism of generating spatial structures is free from the restrictions of the Turing mechanism [], which requires a large difference in diffusion coefficients of the two species involved. Accordingly one can expect DIFICI to be found widely in population dynamics [] and biological morphogenesis [0]. Recently we conducted experiments to study the effect of a differential flow in a biological system, namely quasi one-dimensional colonies of the signaling amoebae Dictyostelium discoideum (D. d.) []. The aggregation of D. d. amoebae after nutrient deprivation is one of the best model systems for the study of spatial-temporal pattern formation at the multicellular level. Starved D.d. cells aggregate to form a multicellular structure by directional movement of cells in response to concentration changes of the chemoattractant cyclic adenosine monophosphate (camp) [,, ]. D. d. cells can be excitable [, ], and when stimulated with extracellular camp, they produce more camp, which in turn stimulates neighboring cells [, ]. This excitable response explains the wavelike pattern of camp, which propagates in the form of either concentric or spiral waves from the self-organized aggregation centers. In the natural environment, this aggregation occurs in forest soil and can be subject to water flow. In general, this medium into which the camp is released is a moving fluid while the cells are crawling on a solid substrate and are not transported by the flow. In our recent experiments with chemotactically active populations of D.d. cells [], a narrow straight microfluidic channel was used and cells were allowed to settle on the glass substrate before a laminar flow of buffer solution through the channel was started. This flow was not strong enough to detach the cells from the substrate and extracellular camp molecules were advected downstream. This differential transport in the extracellular medium induced macroscopic wave trains that developed spontaneously, propagated in the flow direction and had a unique period. In this work, we have investigated the mechanisms of flow-driven waves using the well-established two and three component Martiel-Goldbeter (MG) model []. In the linear regime, our analytical calculations show that the convective transport of extracellular camp in a uniform field of signal-relaying cells leads to a flow-induced instability of the traveling-wave type. In the nonlinear regime, numerical simulations show propagating waves initiated by a convective instability, in agreement with the predictions of the linear analysis.. The Martiel-Goldbeter model The kinetics of the camp production in well-stirred suspensions of D. d. cells are well described by the MG model [], which is based on the relevant kinetic rate laws and the interaction between camp and its membrane receptor (see Fig. ). In this model, extracellular camp binds to the receptor, and the receptor-camp complex activates adenylate cyclase which is an enzyme that produces intracellular camp from ATP. The newly synthesized camp is excreted into the extracellular medium where it binds to

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6 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R Page of 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum with f (γ) = +κγ +γ, f (γ) = L +κl cγ, +cγ Φ(ρ,γ) = λ +Y λ +Y, Y = ργ +γ. where t is equal to the real time multiplicated by k, ρ is the fraction of receptors in the active state, β =[camp] intracellular /K R and γ =[camp] extracellular /K R are dimensionless concentrations of intracellular and extracellular camp, respectively. Here K R = 0 M is the dissociation constant of camp-receptor complex in an active state and k = 0.0 min is the desensitization rate of active receptors. The dimensionless parameters are ǫ = k k e, ǫ = k k i +k t, κ = k k, s = qσ k t+k i α +α, s = kt k eh, λ = λθ ǫ, λ = +αθ ǫ(+α), L = k L = k k and c = K R K D. The values of model parameters are k =. min, k t = 0. min, k e =.. min, k i =. min, σ = 0.0. min, L = 0,L = 0.00, α =, θ = 0.0, λ = 0.0, ǫ =, h = 00, c = 0, and q = 00. These parameter values are the experimental data listed in the last column of Table II in Ref. [], except for the values of k = 0.0 min and k =. min which are adjusted to obtain waves with a period of approximately min. Here σ corresponds to the maximum activity of adenylate cyclase, which is the enzyme producing intracellular camp after being activated by an active camp-receptor complex, and k e is the degradation rate of extracellular camp by the enzyme phosphodiestrase. Lastly, h is the ratio of the extracellular to intracellular volume which varies according to the cell density. For the definition of the other parameters see Fig. and Ref. []. Tyson and coworkers extended the MG model to a field of stationary signaling cells by adding spatial diffusion of camp through the extracellular medium []. The changes in both intracellular camp and the membrane receptors occur only by reactions described by eqns. a,b. However, in the presence of an external flow, the extracellular camp spreads out in the extracellular medium by both molecular diffusion and advection. The reaction-diffusion-advection form of the eqn. c reads k, ǫ t γ = s β γ +ǫ γ +ǫ v γ. () and are the one-dimensional Laplacian and gradient operator for dimensionless spatial variable x defined by x = k ked x from the dimensional space variable x. D = 0.0 mm min [] is the diffusion coefficient of camp and v is the dimensionless flow velocity v = V f / k e D where V f is the dimensional velocity. Here we assume that theflowvelocityv f iswellbelowthecriticalvelocitiesneededtodetachthecellsfromthe substrate. Therefore the cells are not transported with the flow and only extracellular camp is advected downstream. Extracellular free phosphodiestrase is also subjected to the flow and therefore one would expect it to be spatially inhomogeneous or have insufficient concentration. Nevertheless, it is known that phosphodiestrase also occurs in membrane-bound form [,, ] and thus will be active also in the presence of an external flow.

7 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum. Linear stability analysis and numerical simulations of the two-component MG model To obtain a detailed understanding of flow-induced instabilities, we analyze the MG model both analytically and by numerical simulations. The analysis of the MG model is simplified if we eliminate the equation for β (intracellular camp) in favor of ρ (receptor) and γ (extracellular camp) [, ]. In the limit of ǫ ǫ, this simplification is possible which gives β s Φ(ρ,γ). Then eqns. in the presence of diffusion and external flow become t γ = ǫ γ +v γ + ǫ [sφ(ρ,γ) γ], t ρ = f (γ)ρ+f (γ)( ρ), where s = s s. We linearize eqn. about the steady state (γ 0,ρ 0 ) writing and assuming that γ and ρ are small. This gives where and (a) (b) γ = γ 0 +γ, ρ = ρ 0 +ρ, () t γ = a γ +a ρ +ǫ γ +v x γ t ρ = a γ +a ρ, (a) (b) a = sn 0, ǫ a = sm 0, ǫ (a) a = f (γ 0 ) f (γ 0 ), a = ( ρ 0 )A 0 B 0 ρ 0, (b) M 0 = ρ 0γ0(λ λ ) (+γ 0 ) (λ +Y0 ), N 0 = ρ 0γ 0 (λ λ ) (+γ 0 ) (λ +Y0 ), Y 0 = ρ 0γ 0, (a) +γ 0 A 0 = (κl L )c (+cγ 0 ), B 0 = κ (+γ 0 ). Note that the stability of the system in the absence of diffusion and flow requires that the trace of system () denoted by T = a +a and its determinant = a a a a should satisfy the following conditions (b) T < 0, > 0. () We represent the perturbation γ and ρ by the spatial Fourier expansion (γ,ρ ) = π (γ k,ρ k )e ikx+ω(k)t dk (0)

8 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R Page of 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum where ω satisfies the dispersion relation ω +ω( T +ǫ k +ivk)+ a (ǫ k +ivk) = 0. () Note that diffusion alone cannot destabilize the system, because for v = 0, conditions () and a = f (γ 0 ) f (γ 0 ) < 0 imply that Re(ω) is negative for all wavenumbers k. However, a reaction-diffusion-flow system may be unstable for nonzero v. To investigate this flow-driven instability, we first plot Re(ω) as a function of wavenumber for different flow velocities V f (mm/min) = v k e D. If a > 0, then Re(ω) becomes positive at sufficiently large k and the system becomes unstable to short-wavelength perturbations. However, the diffusion term produces a short-wavelength cutoff in the dispersion curves as shown in Fig. a. This entails the appearance of a threshold flow velocity v min, below which the homogeneous steady state is always stable. To obtain the critical velocity, we look at the neutral curve in the v k plane on which Re(ω) = 0. This curve is obtained from the dispersion relation () and is given by v = ( a +ǫ k )(ǫ k T). () k (a ǫ k ) Note that eqn. needs the additional condition a > 0 to be defined on the interval 0 < ǫ k < a. A typical shape of the neutral curve is presented in Fig. b. It is possible to obtain a critical velocity as well as a critical wave number from the conditions Re[ω(k c )] = 0 and d[reω(k c )]/dk = 0. A rough estimate gives k c ( a +a /ǫ ) / and v min [(a +a ) ǫ /a a ] /. Note that v min depends on all the parameters in the model. Exemplarily, Fig. c shows that the threshold velocity V min is smaller for low cell densities (high h values). The comparison between linear stability analysis and numerical simulations is presented in Fig. d, where the values of k with maximum growth rate are plotted for different flow velocities. Interestingly, there is a good agreement between simulations and linear stability analysis especially at large and small values of flow rates. As shown by the bifurcation diagram in Fig., the parameters σ, k e and h are among the important factors governing the dynamics of the MG model (eqns. ). The region of possible convective instability is labelled CU in the figure where the system can sustain convective instabilities for flow velocities above the critical v min. The results in Fig. a predict that convective instabilities can occur for both large and small values of h which correspond to lower and higher cell densities, respectively. However, the window for convective instabilities is slightly narrower for larger values of h (low cell density). More importantly, Fig. b predicts that, for fixed values of the other parameters, there is a finite range of degradation rate k e just before the absolute unstable (AU) regime, where convective instabilities can occur. This window increases slightly as σ increases (i.e. higher activity of the enzyme adenylate cyclase). We solve eqs. numerically using the Merson modification of the Runge-Kutta method with automatic regulation of the time step. The spatial domain consists of 00 grid points in the x direction with a mesh size of x = 0.0 mm. The flow is from

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16 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R Page of 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum to the zero-flux nullcline and the system is at the onset of instability (Fig. b). Above this critical velocity there is no intersection point anymore, except at very small values of camp concentration and a stable limit cycle appears which leads to oscillations in camp concentration (see Fig. c). In other words, close to the inlet, there is a competition between the destabilizing effect of the inlet and stabilizing effect of diffusion and advection. However, there is an upper limit of flow velocity above which the limit cycle disappears and the system is attracted by the fixed point with small concentration of camp (Fig. d). For the set of parameters in Fig. a, SDOs are observed for flow velocities larger than 0. mm/min but smaller than. mm/min. The amplitude of these SDOs decay as the distance to the inlet region increases. Far away from the inlet, the net flux of camp is very close to zero and the new fixed point is stable. In Fig. b, the SDOs are combined with flow driven waves, where the flow velocity is well above the critical velocity v c 0. mm/min. Similarly, a combination of SDOs and flow-driven waves is observed in Fig. c, which is very similar to the experimental observations in Ref. []. In the experiments, the channel length was only mm. Thus Fig. c predicts that for experiments with much longer channels, uniform bulk oscillations may be observed at the downstream of the channel.. Linear stability analysis and numerical simulations of the three-component MG model In this section, we consider the full set of equations with diffusion and advection (eqns. a, b and ). If we assume that the system has a stable steady state (γ 0,ρ 0,β 0 ), then the evolution of a small perturbation is governed by the linearized equations where t γ = a γ +a ρ +a β +ǫ γ v x γ, t ρ = a γ +a ρ +a β, t β = a γ +a ρ +a β, (a) (b) (c) γ = γ 0 +γ, ρ = ρ 0 +ρ, β = β 0 +β, () where γ,ρ,β are small and a ij are the elements of the Jacobian matrix a = ǫ, a = 0, a = s ǫ, (a) a = ( ρ 0 )A 0 B 0 ρ 0, a = f (γ 0 ) f (γ 0 ), a = 0, (b) a = s N ǫ, a = s M ǫ, a = ǫ, (c) with M 0, N 0, A 0 and B 0 defined in Eqn.. We first consider the system without diffusion and convection and introduce the following definitions: T = a +a +a is the trace of the system, is its determinant, T ij = a ii +a jj (i < j) is the trace of the subsystem formed by γ,ρ (β = 0) or γ,β (ρ = 0)

17 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum orρ,β (γ = 0),and ij = a ii a jj a ij a ji (i < j)isthedeterminantofthesamesubsystem, while Σ = + +. Based on the Hurwitz criterion [], the stability of the entire three variable system requires that T < 0, < 0, and TΣ < 0. () Let us now consider the system () with flow but without the diffusion term. We represent the perturbations (γ,ρ,β ) by the spatial Fourier transform (γ,ρ,β ) = (γ k,ρ k,β k )e ikx+ω(k)t dk. The equations for the Fourier components then become π t γ k = a γ k +a ρ k +a β k vikγ k, t ρ k = a γ k +a ρ k +a β k, t β k = a γ k +a ρ k +a β k, (a) (b) (c) We form the characteristic polynomial P(iω) = P R (ω)+ip I (ω) where ω, P R, and P I are all real. The real and imaginary parts of P(iω) are P R (ω) = Tω +vkt ω, P I (ω) = ω vkω +Σω +kv For convective instability to occur, the following conditions should be fulfilled [] (a) (b) b/a > 0 and b a > 0 () where b = +a T ( + ). Including the diffusion of extracellular camp in addition to the convection term generates a short wavelength cutoff in the dispersion relation and leads to the appearance of a critical velocity, similar to the two-component case. We used the criteria in eqn. to explore the regions of convective instabilities in the k e h and σ k e parametric planes (see Fig. ). Note that again, similar to the two-component MG model, regions of CU occur right before the absolute instability (AU) but in a narrower region (Fig. b). We extended the numerical simulations described in Sec. to the three-component MG model and performed simulations for the set of parameters inside the CU regime, namely k e =. min and σ = 0. min. This point is marked by the symbol in Fig. b and is located close to the transition line but in the convectively unstable region of the phase diagram. Note that, in the absence of flow (and diffusion), the homogeneous steady state is stable to small perturbations; however an external flow above the threshold velocity v min may lead to convective instability. The corresponding wave pattern is illustrated in Fig. 0a, c. Remarkably, the flow driven instability waves of camp need more time to be fully developed. Since the system is very close to the transition line to the stable region, the wave fronts have a lower concentration of camp. However, deep inside the CU regime and close to the lower transition line in Fig. b, the instability waves appear earlier in time and have higher concentrations of camp (see Fig. 0b, d). This point is denoted by + in Fig. b and corresponds to parameter values k e =. min and σ = 0. min.

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20 CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R Page of 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum signaling, where the cell population is assumed to have a uniform distribution. However as the cells start to aggregate, variations in the cell density become significant. Future studies should attempt to include chemotactic cell movement in the simulations to study the distribution of cell population in the presence of flow-driven waves. Our results may also provide a starting point for the study of pattern formation and locomotion-driven instabilities in other biological systems.. Acknowledgment We are grateful to Alan Pumir for discussions and comments. A. G. was supported by Dorothea-Schlözer scholarship from Georg-August University of Göttingen, Germany. O. St. acknowledges support provided by the A.v.Humboldt Foundation and the National Science Foundation (CHE-). References [] Rovinsky A B and Menzinger M Phys. Rev. Lett. [] Rovinsky A B and Menzinger M Phys. Rev. Lett. [] Rovinsky A B and Menzinger M Phys. Rev. Lett. 0 [] Toth R, Papp A, Gaspar V, Merkin J H, Scott S K and Taylor A F 0 Phys. Chem. Chem. Phys., [] Malchow H 00 J. Theoret. Biology [] Malchow H 00 Freshwater Biology [] Scheffer M J. Plankt. Res. [] Turing A Philos. Trans. R. Soc. London Ser. B [] Malchow H Ecological Modelling [0] Kaern M, Menzinger M, Satnoianub R, and Hunding A 0 Faraday Discussions [] Gholami A, Steinbock O, Zykov V and Bodenschatz E Phys. Rev. Lett., 00 [] Gerisch G Annu. Rev. Biochem. [] Devreotes P N Science 0 [] Parent C A and Devreotes P N Annu. Rev. Biochem. [] Holden A V, Markus M and Othmer H G Nonlinear Wave Processes in Excitable Media (Plenum Press, New York) [] Murray J D Mathematical Biology (Springer-Verlag, Berlin) [] Martiel J L and Goldbeter A Biophys. J. 0 [] Pupillo M, Insall R, Pitt G S, and Devreotes P N Mol. Biol. Cell. [] Saran S, Meima M E, Alvarez-Curto E, Weening K E, Rozen D E, Schaap P 0 J. Muscle Res. Cell Motil. [] Brown S S and Rutherford C L 0 Differentiation [] Tyson J J, Alexander K A, Manoranjan V S and Murray J D Physica D [] Chang Y Y Science, [] Shapiro R I, Franke J, Luna E J, Kessin R H Biochim. Biophys. Acta, [] Bader S, Kortholt A., Hastert P J M V 0 Biochem. J., [] Decave E, Garrivier D, Breche Y, Fourcade B, and Bruckert F 0 Biophy. J., [] Wessels D, Brincks R, Kuhl S, Stepanovic V, Daniels KJ, et al 0 Eukaryot Cell, [] Meinhardt H Models of Biological Pattern Formation (Academic Press, New York) [] Warren A J, Warren W D, and Cox E C Genetics, [] Whitesides G M, Ostuni E, Takayama S, Jiang X, Ingber D E 0,

21 Page of CONFIDENTIAL - AUTHOR SUBMITTED MANUSCRIPT NJP-0.R 0 Flow-driven instabilities during pattern formation of Dictyostelium discoideum [] Gross J D, Peacey M J, and Trevan D J J. Cell Sci., [] Devreotes P N, Potel M J and MacKay S A Dev. Biol. [] Lindner J, Sevcikova H, and Marek M 0 Phys. Rev. E 0 [] Korn G A and Korn T M Mathematical Handbook (McGraw-Hill, New York) [] Garcia J M and Neufeld Z 0 Phys. Rev. E 0 0 [] Lauzeral J, Halloy J, Goldbeter A Proc. Natl. Acad. Sci. [] Palsson E and Cox E C Proc. Natl. Acad. Sci.