Stochastic Dynamics Out of Equilibrium @ IHP 11/04/2017 Monte Carlo method for kinetic chemotaxis model and its applications on traveling pulse and pattern formation Graduate School of Simulation Studies, University of Hyogo Shugo YASUDA In collaboration with Benoît Perthame Laboratoire Jacques-Louis Lions
Plan of Talk 1. General Introduction 2. Kinetic Chemotaxis Model 3. Monte Carlo Method 4. Application 1: Traveling pulse 5. Application 2: Pattern formation 6. Concluding remarks 2
Introduction Run-and-Tumble Bacteria Collective dynamics of bacteria Pattern formation by Budrene and Berg, Nature 349, 630 (1991). E. Coli 2~4 µ m “Run”: Flagella rotate counterclockwise “Tumble”: flagella rotate clockwise Homepage of H. C. Berg Bacteria communicate via chemical cues http://www.rowland.harvard.edu/labs/bacteria/
Motivation • Multiscale mechanism and mathematical hierarchy in the collective dynamics of bacteria. – Relation between macroscopic phenomena, individual motions, and internal states • Simulation method – Extensible (modeling) and Scalable (computation) • Applications – Traveling pulse, Pattern formations, .... 4
Objective of study • Development of a Monte Carlo method for chemotactic bacteria based on a kinetic chemotaxis model . • Applications on traveling pulse and pattern formation. – Validity of the MC method via comparisons to the theoretical and experimental results. – A new theoretical result on the instability analysis of a kinetic chemotaxis equation. 5
Why kinetic model? • Mesoscopic modeling involving the individual dynamics (multiscale nature) – H. G. Othmer, S. R. Dunbar, and W. Alt (1988); R. Erban and H. G. Othmer (2004); Y. Dolak and C. Schmeiser (2005); N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler (2007); etc.. • Mathematical hierarchy – T. Hillen and H. G. Othmer (2000), (2002); F. A.C.C. Chalub, P. Markowich, B. Perthame, and C. Schmeiser (2004); F. James and N. Vauchelet (2013); G. Si, M. Tang, and X. Yang (2014); B. Perthame, M. Tang, and N. Vauchelet (2016). • Development of experimental technologies – J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin, and P. Silberzan (2011); C. Emako, C. Gayrard, A. Buguin, L. Almeida, and N. Vauchelet (2016). 6
Plan of Talk 1. Introduction 2. Kinetic Chemotaxis Model 3. Monte Carlo Simulation 4. Application 1: Traveling pulse 5. Application 2: Pattern formation 6. Concluding remarks 7
Schematic of kinetic modeling Biased random motions searching for the chemical attractants bacterium 2~4 µ m Chemical cues Foods Secretions << 1 µ m 8
Schematic of kinetic modeling Continuum description for chemical cues, 𝑇(𝑢, 𝒚) Stiff response ramp up Kinetic description for bacterial density ramp down with the velocity distribution function 𝑔(𝑢, 𝒚, 𝒘) From Block SM, Segall JE, Berg HC, J. Bacteriol 154, 312 (1983) 9
Individual Motions of Bacteria Run-and-Tumble motion Stochastic process e.g., E-coli 1. Tumbling at some rate 𝜇 . 2. Reorientation followed by some PDF 𝐿(𝑤, 𝑤 - ) . 3. Cell division/extinction with some rate 𝑠 . Bacterial density 𝑔(𝑢, 𝑦, 𝑤) changes during the stochastic process. Homepage of H. C. Berg http://www.rowland.harvard.edu/labs/bacteria/ 10
� � � � � � Kinetic Chemotaxis model with growth term Cell division 𝜖 1 𝑔 + 𝑤 3 𝜖 4 𝑔 = 6 𝑈 𝑤, 𝑤 - 𝑔 𝑢, 𝑦, 𝑤 - 𝑒𝑤 - − 6 𝑈 𝑤 - , 𝑤 𝑔 𝑢, 𝑦, 𝑤 𝑒𝑤 - + 𝑠𝑔(𝑢, 𝑦, 𝑤) Gain Term Lost Term Transient kernel 𝑈 𝑤, 𝑤 - = 𝜇 𝑤 - 𝐿(𝑤, 𝑤 - ) 𝑤′ 6 𝐿 𝑤, 𝑤 - 𝑒𝑤 = 1 Searching for foods and chemical cues along their trajectory 11
Scattering Kernel • Tumbling rate Temporal variation along the trajectory 𝜇 𝑤 - = 1 𝐸 log 𝑂 𝐸 log 𝑇 2 𝜔 ? E + 𝜔 G E 𝐸𝑢 𝐸𝑢 F- F- – Stiff response function Nutrient-Poor 𝜔 𝑌 = 𝜔 I − 𝜓 tanh 𝑌 𝜀 Nutrient-Rich • Mean tumbling rate 𝜔 I • Modulation parameter 𝜓 G,? • Stiffness parameter 𝜀 PQ 12
Scattering Kernel • Reorientation (e.g., von Mises distribution) exp − 1 − cos 𝜄 𝜏 a 𝐿 𝒘, 𝒘 - = 𝒘 a 𝜏 a 1 − 𝑓 P a 2𝜌𝑊 d W I 𝜄 𝒘′ – Reorientation angle q – Constant Speed 𝒘 = 𝑊 I . – Standard deviation s Q • Uniform scattering 𝐿 = W as 𝜏 → ∞ . STU V
Basic equations • Kinetic chemotaxis | e 0 | = 1 – Modulation of tumbling frequency, for example, – PDF of reorientation angle, for example, ⇣ ⌘ e 0 1 � ˆ e · ˆ exp σ 2 ˆ e 0 ) = K (ˆ e , ˆ ⇣ ⌘ 1 − e � 2 (von Mises distribution) 2 πσ 2 σ 2 14
Basic equations • Reaction-Diffusion equations of chemical cues x ) = 1 Z ˆ ρ (ˆ f (ˆ e 0 ) d Ω (ˆ e 0 ) t, ˆ t, ˆ x , ˆ ˆ 4 π | ˆ e 0 | =1 15
Parameters • Mean run length (or Knudsen number) • Stiffness and modulation in response function • Other Parameters 16
Plan of Talk 1. Introduction 2. Kinetic Chemotaxis Model 3. Monte Carlo method 4. Application 1: Traveling wave 5. Application 2: Pattern formation 6. Concluding remarks 17
Simulation method • Monte Carlo method for chemotactic bacteria coupled with a finite volume scheme for chemical cues. • Motions of bacteria calculated by MC particles. • Macroscopic quantities are calculated based on a lattice-mesh system. • Similar to the DSMC method for the Boltzmann equation of gases. 18
Lattice System and MC Particles • Motions of bacteria by Monte Carlo particles • Macroscopic quantities on a lattice-mesh system non-flux non-flux Periodic ∂ x S = ∂ x N = 0 ∂ x S = ∂ x N = 0 Specular r n ˆ ( l ) Specular i ˆ ˆ ˆ ˆ ˆ ˆ x 0 x 1 x n x n +1 x I x − 1 x I x i + 1 y superscript n : time step, subscript i : lattice site, z and subscript ( l ): index of particle x 19
Calculation of Chemical Cues • Finite volume scheme on the lattice mesh Population densities are calculated from the numbers of MC particles in each lattice site. 20
Monte Carlo Method e 0 . Initialization: Distribute particles according to 𝑔 f 0 (𝒇 i) . 1. Move particles in a time-step size ∆𝑢 . 2. Calculation of local concentration of chemical cues. 3. Tumbling of each particle by a probability e(𝒇 i′ k )Δ𝑢̂ . 𝜇 i - . 4. Reorientation angle by 𝐿 𝒇 i, 𝒇 5. Division by a probalibity 𝑠̂∆𝑢̂ . 6. Return to 1. 21
Monte Carlo Method 0 . Initialization: e I (𝒇 i) . MC particles are distributed according to 𝑔 f I via Calculate particle number in the i th lattice site 𝜈 f – where 𝑥 I is the uniform weight of a single MC particle – – In each lattice site, particles are randomly distributed. e I (𝑓̂)/𝜍 Velocity of particle is determined by the PDF 𝑔 i 𝑗 0 – f 22
Monte Carlo Method 1. Movement: Particles move with their velocities in a time step size ∆𝑢 . 𝒔 i (k) , 𝒇 i (k) : position and velocity of l th particle – Particles beyond the boundaries are removed and new ones are inserted following the boundary conditions. – Count the numbers of simulation particles in each lattice site 𝜈 ftuQ ( 𝑗 = 0, … , 𝐽 4 − 1 ). 23
Monte Carlo Method 2. Calculation of chemical cues at each lattice site. tuQ = 𝑥 I 𝜈 ftuQ /[ 𝑀 0 ∆𝑦 3 𝜍 I ] . – with 𝜍 i f 24
Monte Carlo Method 3. Tumbling of the l th particle by a probability t ) , } 0 ∆𝑢̂ Ψ •(𝒇 i k 𝜔 tuQ − log𝑂 (k) t log𝑂 (k) 𝐸 log 𝑂 E ⟹ 𝐸𝑢 Δ𝑢 𝒇 t → 𝒔 tuQ . Temporal variation along the pathway 𝒔 i (k) i (k) 25
Monte Carlo Method 3. Tumbling of the l th particle by a probability t ) , } 0 ∆𝑢̂ Ψ •(𝒇 i k 𝜔 t , 𝑂 t : sensed by the l th MC particle at 𝒔 e (k) • (k) t 𝑇 i (k) – – Calculated by the interpolation, The particles that stay at the same lattice site after ∆𝑢 – passes can sense the gradients. 26
Monte Carlo Method 4. Reorientation angle by a probability tuQ i (k) 𝒇 t ). •(𝒇 tuQ , 𝒇 i k i (k) 𝐿 𝜄 t 𝒇 i (k) Reorientation angle 𝜄 (for von Mises distribution) – 5. Cell divisions (or deaths) with a probability 𝑠̂∆𝑢̂ . 6. Return to step 1 (Movement Step). 27
Monte Carlo Method • First order accuracy in time and space under the assumption of the law of large numbers. – B. Perthame and S. Yasuda, “Self-organized pattern formation of run-and-tumble chemotactic bacteria: Instability analysis of a kinetic chemotaxis model”, hal-01494963 (2017). 28
Plan of Talk 1. Introduction 2. Kinetic Chemotaxis Model 3. Monte Carlo Simulation 4. Application 1: Traveling pulse 5. Application 2: Pattern formation 6. Concluding remarks 29
• Literature on traveling pulse by J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguinn, and P. Silberzan, PNAS 108, 16235(2011) V wave =4.1 µ m/s
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