Numerical method for osmotic water flow and solute diffusion in moving cells

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1 Numerical method for osmotic water flow and solute diffusion in moving cells Lingxing Yao and Yoichiro Mori Case Western Reserve University and University of Minnesota IMA Workshop Electrohydrodynamics and Electrodiffusion in Material Sciences and Biology March 15, 018 Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

Cell structure and movement 1 3 1 https://www.cellsalive.

2 Cell structure and movement Torsten Wittmann, Scripps Research Institute. Actin filiments (purple), microtubules (yellow), and nuclei (green) 3 A. Schwab, A. Fabian, P. J. Hanley, and C. Stock, Physiol. Rev., 9, , (01) Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, 018 / 5

Osmotic engine experiment Cancer cell can move through confined extracellular matrix, including longitudinal tracks of varying length, with actin polymerization and depolymerization inhibited Osmotic

3 Osmotic engine experiment Cancer cell can move through confined extracellular matrix, including longitudinal tracks of varying length, with actin polymerization and depolymerization inhibited Osmotic shock, or polarized distribution of ion channels and aquaporins can lead to sustained cell migration in confined collagen matrix For cancer cells, actin polymerization cycle and water permeation due to osmosis may be the two mechanisms to drive the migration 1 1 K. M. Stroka, H. Jian, S. Chen, Z. Tong, D. Wirtz, S. Sun, and K. Konstantopoulos, Cell, 157, , (014) Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

4 Our goal and model assumptions We want to verify the role of chemical osmosis in driving cell migration by studying the diffusion of chemicals, osmotic water flow, and actin network, in a model cell system, which includes: Cell membrane is an elastic structure dividing the intra- and extra-cellular space and allows water and chemicals to pass through Chemicals diffuse inside and outside the cell, and cross cell membrane via passive or active channels Difference in solute concentrations across the membrane, together with mechanical forces from the membrane, lead to movement of the membrane and transmembrane water flow Actin network is present over the entire intra-cellular space as another phase and is able to polymerize/depolymerize (currently ongoing with some results) Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

5 Model equation part 1: fluid flow 1 The variables in this module include fluid velocity u, pressure p. We have incompressible Stokes (or Navier-Stokes) equation: u t + u u = Σ m(u, p), u = 0, where ν is fluid viscosity, and viscous stress Σ m (u, p) ν( u + ( u) T ) pi. Fluid velocity is continuous across membrane: [u] = 0 The membrane may not be moving with fluid velocity: u X t = f w n, with water flux f w = k w [ψ], and water potential ψ = c (Σ m (u, p)n) n. Here c is the chemical concentration in fluid. 1 L. Yao and Y. Mori, A numerical method for osmotic water flow and solute diffusion with deformable membrane boundaries in two spatial dimension, Journal of Computational Physics, 350, , 017 Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

6 Model equation part : membrane elasticity and force The membrane is denoted by X(s) (Lagrangian description of the membrane), which has elastic forces: (( ) ) X F elas = k elas s s l τ, τ = X 1 X s s, F bend = k bend 4 X s 4, Along cell membrane, viscous stress difference is balanced by the mechanical force of the membrane (n is outward normal along the membrane): [Σ m (u, p)n] = F mem X s 1, F mem = F elas + F bend Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

7 Model equation part 3: chemical concentration The chemical is defined over entire computational domain, with concentration c: c + (uc) = (D c), t D is diffusion coefficient. Along cell membrane, chemical boundary condition satisfies: (cu D c) n = c X t n + f c + f p, on Γ I, Γ E, where f c = k c [c] is transmembrane passive chemical flux, and f p is active flux H(s, c i, c e ) = f p = k p H(s, c i, c e ), k p = k p (s) { c i if k p (s) 0, c e if k p (s) < 0. X s 1, Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

8 The free energy identity 1 The model proposed so far is thermodynamically consistent, as it has the following energy identity (for Stokes flow): d dt (E bulk + E mem ) = I J, E bulk = ωdx, ω = RT (c ln c c), Ω i Ω e ( ( ) X E mem = k elas Γ ref s l X ) + k bend s ds, I = (ν S u + DRT ) c µ dx, µ = RT ln c, Ω i Ω e J = ([ψ]f w + [µ](f c + f p )) dm Γ, dm Γ = X s ds. Γ The entropic and membrane elastic free energy are dissipated through bulk flux (viscous and solute dissipation) and membrane flux (water and chemical flux). 1 Based on Y. Mori,, C. Liu, and R. S. Eisenberg, Physica D, 40, , (011) Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

9 The design of numerical method Fluid equations are rewritten into immersed boundary (IB) form: u t where δ is Dirac delta function. + u u = ν u p + f, u = 0, f(x, t) = F mem (X)δ(x X(s, t))ds, Γ Membrane location X is updated semi-implicitly using water permeable velocity boundary condition in the IB framework Chemical concentrations c ij are defined at all cell centers over a fixed computational grid, and updated with a Cartesian grid method. Chemical concentrations c b i and cb e are also defined along Γ i and Γ e to enforce chemical boundary condition, and increase numerical stability. Fluid structure interaction and chemicals update are decoupled on the cell membrane using fractional steps in time. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

10 Diagram of computational grids and variables Ω 1 Γ 1 Ω 3 Γ 3 Ω Γ Ω 0 u = (u, v), p are in MAC arrangement, X i, i = 1,..., n ring are IB points to discretize cell membrane Γ, chemical on computational grid c ij are at all computational cell centers, and c b i,e are defined at crossings of the fixed Eulerian grid and Γ. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

11 Numerical algorithm 1 Given u n, v n, p n and X n, use explicit IB method to compute u n+1, v n+1 and p n+1. Once this is found, use X n, c b,n i,e as well as the newly found u n+1, v n+1 to get the new IB locations X n+1 : f n+1 w X n+1 = X n + (u n+1 fw n+1 n) t = k w [ψ w ] = k w ( [c] n F mem (X n+1 ) Xn s 1 n) Given X n+1 i and u n+1, update chemical c n to c n+1 at all cell centers by using Cartesian grid method (need modification near membrane) c n+1 c n + (u n+1 c n+1 ) = D c n+1 t with chemical boundary condition, which is enforced with help from c b,n+1 i,e fw n+1 c D c n = k c [c] + f p Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

12 Numerical details for IB step: fluid equations u n+1 u n i,j+ 1 i,j+ 1 +[(u t i,j+ 1 C x + E x E y v i,j+ 1 C y)u i,j+ 1 ]n+ 1 = ν L(un+1 + u n i,j+ 1 i,j+ ) D 1 x p n+1 + f n i+ 1,j+ 1 x,i,j+, 1 v n+1 v n i,j+ 1 i,j+ 1 +[(E x E y u t i+ 1,j C x + v i+ 1,j C y )v i+ 1,j ] n+ 1 = ν L(vn+1 + i+ 1,j vn i+ 1,j) D y p n+1 + f n i+ 1,j+ 1 y,i+ 1,j, 0 = Dx u i+1,j+ 1 + D y v i+ 1,j+1. D ± x w α,β = ± w α±1,β w α,β h, D y ± w α,β = ± w α,β±1 w α,β, h Lw α,β = w α+1,β + w α,β+1 + w α 1,β + w α,β 1 4w α,β h, E x w α,β = w α+ 1,β + w α 1,β, C x w α,β = w α+1,β w α 1,β h Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

13 Continue IB step: Elastic force and Delta function f n x,i,j+ 1 = n ring k=1 n ring f n y,i+ 1,j = k=1 where delta function is given by F n x,kδ h (x i X n k )δ h (y j+ 1 Y n k ) s F n y,kδ h (x i+ 1 Xn k )δ h (y j Y n k ) s δ h (r) = 1 ( r ) h φ, h 1 8 (3 r r 4r ) r 1, 1 φ(r) = 8 (5 r r 4r ) 1 < r, 0 < r. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

14 Numerical details for chemicals: auxiliary variables Extrapolations are used to ensure chemicals are updated using chemicals from proper side: c p1 = (1 θ) + θ c 3(1 θ) p θ c 6 p 5 + (1 + θ)( + θ) cb i p p1 va (xp, yp) n (x b 1, y1) b C p3 B A θ y vo (x b, y) b p4 (X n, Y n ): IB point (Marker); (x b 1, y b 1)& (x b, y b ): grid crossings of interface Γ and lines connecting cell centers between p 1 to A and p 3 to A; A, B, C: irregular cell centers Chemical boundary condition is p7 p5 p6 (Xn+1, Yn+1) (Xn, Yn) (Xn 1, Yn 1) f n+1 w c D c n = k c [c] + f p, where the gradient will be enforced using c b i (v o / v o ) 3 v o cb i + v o c B 1 v o [(1 θ)c C+θc p7 ], (v o is off cell center direction that n is decomposed into) Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

15 Numerical details of freshly cleared chemicals At freshly cleared point F (on extracellular side) is replaced by c F t c n+1 F cn+1 F c n e,p F t=n t t c n F, t ũ F D 0 xc n+1 F ṽ F D 0 yc n+1 F, t n t n+1 p1 py pf F px p where with D 0 x y w = 1 (ũ F, ṽ F ) = (D +x y w + D x y w ), ( xf x F t, y F y F t ). Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

16 Test of Cartesian grid method for updating chemicals Computational domain [0, 1] [0, 1] with initial membrane in circle: X(s) = cos s, sin s, and chemical field: c(x, y, 0) = 0.5(1.5 + sin(π(x 0.5)))(1 + sin(π(y 0.5))) The fluid velocity and cell motion are prescribed by u = u(x, y), v(x, y) = 1 4 (y 1 ), 0 ; dx dt = ( 1 4 (y 1 ) ) 1 cos t, 0 Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

17 Snapshots of chemicals and cell locations t=0.00 t=0.50 t=1.00 t=1.50 t=.00 t= Chemical Field Chemical Field Chemical Field x t =0.05 t =0.5 Test chemical module t =.5 Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

18 Convergence results of in chemical module t e I L e I L e E L e E L ɛ { x, t}/ /ɛ { x, t}/ ɛ { x, t}/ /ɛ { x, t}/ ɛ { x, t}/ /ɛ { x, t}/ ɛ { x, t}/ /ɛ { x, t}/ ɛ { x, t}/ /ɛ { x, t}/ At different times, convergence rates of chemical c on cell centers for the analytical IB locations simulation. x = 1/64, and t = Superscript I and E indicates the intracellular and extracellular cell centers, and subscript L and L indicate errors norms used. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

19 π Test of full coupling of fluid flow and chemicals Computational domain [0, 1] [0, 1] with initial membrane configuration X(s) = cos s, sin s, and chemical field is constant 1 everywhere. Parameters are listed here k c k w D k elas k bend l r ν Case 1a kp(s) 0 3π π π s with active pump strength: twid = 0.5hwid 1 s k p (s) = k h (e (h wid ) + e (s π) (h wid ) ) + k t e (s π) where k h = k t =, h wid = t wid = 0.π, and s [0, π] (t wid ), Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

20 Velocity and chemical snapshots of full coupling Velocity Chemical Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

21 Convergence results for all variables Rate on variables t = 0.5 t = 1.5 t =.5 t = 3.75 ɛ(c E ) { x, t}/ / ɛ(c E ) { x, t}/ ɛ(c I ) { x, t}/ / ɛ(c I ) { x, t}/ ɛ(c L E ) { x, t}/ / ɛ(c L E ) { x, t}/ ɛ(c L I ) { x, t}/ / ɛ(c L I ) { x, t}/ ɛ(x) { x, t}/ / ɛ(x) { x, t}/ ɛ(u) { x, t}/ / ɛ(u) { x, t}/ At different times, convergence rates of chemicals at cell centers on intracellular side c I and extracellular side c E of Γ, chemicals at IB points on intracelular side c L I side and extracellular side cl E of Γ, and locations of IB points X are recorded. Here x = 1/64, t = 0.005, and n ring = 160 for this pair of x and t. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

22 Full model predictions with varying membrane elasticities t=0.0 t=3. t=6.4 t= t=0.0 t=3. t=6.4 t= t=0.0 t=3. t=6.4 t= t=0.0 t=3. t=6.4 t= Comparison of cell development with different membrane elasticity while all other parameters and pump are the same. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, 018 / 5

23 Full model predictions in permeability and chemical flux speed kc = 0.1, D = 0.1 kc = 0., D = 0. kc = 0.4, D = 0.4 speed kw = 1.11, D = 0.1 kw = 1.11, D = 0. kw = 1.11, D = 0.4 kw =., D = 0.1 kw =., D = 0. kw =., D = kw kc 1 Speed is measure by using the distance traveled by the center of mass for the cell membrane. 1 Left panel: the speed (unit µm/s) calculated to t = 40s, vs water flux coefficient k w (unit µm s/kg). Right panel: the speed calculated to t = 40s vs water passive chemical flux constant k c (unit µm/s), at different diffusion coefficient D (unit 10 3 µm /s) and k w. Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

24 Conclusions, current and future work Conclusions: We construct a thermodynamically consistent model for osmotic water flow and cell movement, and design a numerical method that is capable of handling fluid structure interactions and chemical osmosis and their coupling. Fluid structure interaction is computed using IB method with semi-implicit treatment on advancing IB points Chemical update is obtained by using a Cartesian grid method for advection diffusions in moving domains, with special cares given to enforce boundary condition and deal with freshly cleared point test cases Current and future work Include new phase of actin network inside the cell and adapt our numerical scheme to handle the full coupling of chemicals, network, and fluid flow (done implementation, testing with applications) Include charges of chemicals and electro diffusion into our model and numerical framework Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

25 Acknowledgement Yizeng Li, Sean Sun Johns Hopkins University Alex Mogilner New York University Aaron Fogelson University of Utah Funding support Yoichiro Mori: NSF DMS Lingxing Yao: NSF DMS Thank you! Any questions? Yao, Mori (CWRU, UMN) Simulating osmotic flow in cells IMA, March 15, / 5

Introduction to immersed boundary method

Introduction to immersed boundary method Ming-Chih Lai mclai@math.nctu.edu.tw Department of Applied Mathematics Center of Mathematical Modeling and Scientific Computing National Chiao Tung University 1001,