Hierarchy of PDE Models of Cell Motility · waves and sharp interface limit, Comptes Rendus...

Hierarchy of PDE Models of Cell Motility

L. Berlyand, Penn State University, USA

Collaborators:

PSU postdocs: J. Fuhrmann, M. PotomkinV. Rybalko (ILTPE)

Phase-field model of cell motility: travelingwaves and sharp interface limit,Comptes Rendus Mathematique (2016)with M. Potomkin, V. Rybalko

Bifurcation of traveling waves in aKeller-Segel type free boundary modelof cell motility,with J. Fuhrmann, V. Rybalko

November 25, 2017

Supported by:NSF grant DMS-1106666

L. Berlyand Hierarchy of PDE Models of Cell Motility

Outline of the talk

I Biology/physics of crawling cell motion

I Phase-field (PF) model of crawling cell:I reduction of complexity via sharp interface limit eqn.;I standing & bifurcation to traveling waves & hysteresis

I Free boundary model of crawling cell

I existence of standing waves and spectral analysis of linearizedoperator

I bifurcation of traveling waves from standing waves

I Explaining key bio phenomena:I steady motility via traveling waves, symmetry breaking via

instabilitiesI new critical parameter value found

Previously: models studied numerically; our work: analysis.


Mathematics of Active Materials

Cell motility: ability to move spontaneously via consumption ofenergy from internal sources or from the environment− example for a new class of problems coming from biology andstatistical physics of out-of-equilibrium systems that describe socalled active matter.

Active materials consume energy from environment and alter theproperties of the surrounding, e.g., swimming bacteria, motile cells,bird flocks, fish schools.

Self-propulsion − mechanism to convert chemical energy tomechanical.

Striking novel properties not available in equilibrium systems⇒ promising applications require new mathematical tools.


Experimental results on keratocyte motion

Different modes of cell motion depending on bio and physical parameters

Motion with constant shapeSource: E.L. Barnhart, et al.

Wobbling (bipedal) cell motionSource: J. A. Theriot


Motile Cells, Example of Active Material

Keratocyte cells (e.g., in fish scales & human corneas) - crawling cellsimportant in wound healing

Top: resting cell (symmetric),Bottom: motile cell(asymmetric, movingdownwards)

Ideal for modeling and experiments:

1. Crawl on substrates, pancake shaped⇒ 2D models and easy forexperiments, lots of data available

2. Resting (circular) cells mayspontaneously break symmetry andinitiate motion.

3. Once in motion, keeps traveling withconstant velocity and shape withoutexternal stimuli.


Mechanisms for cell motility

fast polymerizationfast depolymerization myosin (1 step)

growth speed shrinking speed

contraction speed

(pointed end) (barbed end)

contraction speed

two antiparallel actin filaments(red) being pulled along each otherby myosin (blue) ⇒ contraction

Substrate

motion

lamellipodium viewed from the sidewith integrin adhesions (yellow)between actin meshwork andsubstrate

I cell shape and position - drivenby competition of protrusion/contraction (inside) versusmembrane (boundary) curvature

I Actin filaments (long polymers) -polymerize at leading edge oflamellipodium (flat front of cell)⇒ protrude front of cell

I Myosin motors - bind two actinfilaments together ⇒ contractilestress in actin network

I Integrins - attach actin filamentsto substrate ⇒ adhesion sitesyield drag force


Goal of the theoretical studies

Competition: mean curvature motion vs protrusion/adhesion

Goal: explain via math models fundamental biological phenomena:

I Resting cell (modeled by standing waves)

I Initiation of spontaneous motion and breaking of symmetry(modeled by bifurcation from standing to traveling waves)

I Steady motility of the cell in absence of external cues(modeled by traveling waves, c.f. celebrated KPP model inparabolic systems )


Review of some recent PDE models

Free boundary type models studied numericallyI Rubinstein, Jacobson, Mogilner (2005) - introduce 2D eqns: actin,

myosin, elastic membrane. Numerics: steady motion, turning cells.

I Keren, Pincus, Allen, Barnhart, Marriott, Mogilner, Theriot (2008) -cell shape determined by: area, aspect ratio

I Barnhart, Lee, Allen, Theriot, Mogilner (2015) - Experiments andfree boundary model: actin flow, adhesion/myosin for initiation ofmotility, stochastic fluctuations of adhesion.

I Recho, Putelat, Truskinovsky (2015) - 1D PDE model: myosin only,onset of motion and stop (arrest), bifurcation analysis via numerics.

Phase-field modelsI Ziebert, Swaminathan, Aranson (2011) - introduced model coupled

with actin flow. Numerics: initiation, steady motion, stick-slip.

I Camley, Zhao, Li, Levine, Rappel (2013) - Introduced model forpatterned substrates. Numerics: steady motion, turning, oscillations.

•Our study is analytical with numerical examplesL. Berlyand Hierarchy of PDE Models of Cell Motility

Hierarchy of Models of Cell Motility in order of decreasing complexity

I Phase-Field (PF) Model: additional PF parameter; contains mostdetails but highest computational cost; lower the cost by reducingdimension;

I Free Boundary Problem (FBP): Biophysics from first principles.Numerical difficulty: tracking free boundary (unlike in PF model).

I Level Set Model: cell boundary described by level set of auxiliaryfunction ϕ: Γ(t) = {ϕ = 0}, ϕ satisfies PDE. PF captures physicsof cell interior; level set model describes advection of interface only.

I Sharp Interface Limit (SIL) for PF: drastic reduction ofcomplexity, special scaling of PF model.

I Agent Based Model: focus on collective behavior rather thanindividual shape dynamics; e.g., represents each cell by a point;minimal model for individual cell.

In this talk, PF Model with its SIL and FBP are considered.


Part I: Phase-Field ModelReduction of complexity via Sharp Interface Limit

Phase-Field model is a PDE approach used to describe theevolution of an interface between two phases.We study a phase-field model of motion of eukaryotic cell. Themodel consists of two parabolic PDEs with gradient coupling andmass conservation constraint. These features prevent from usingclassical techniques (e.g., vanishing viscosity method andΓ-convergence).


A phase-field model of keratocyte motility (Aranson et al.)

Scalar ρε(x , t): non-physical phase-field parameter (not conserved)Vector Pε(x , t): averaged orientation of filament network

∂ρε∂t

= ∆ρε −1

ε2W ′(ρε)− Pε · ∇ρε + λε(t) in Ω ⊂ R2, (1)

∂Pε∂t

= ε∆Pε −1

εPε − β∇ρε in Ω ⊂ R2. (2)

W (ρ) a potential with two equal wells at 0 and 1, e.g. W (ρ) = ρ2(1− ρ)2(symmetric), W (ρ) = ρ2(1− ρ)2(1 + ρ2) (asymmetric),λε(t) =

´ W ′(ρε)ε2

+ Pε · ∇ρεdx : Lagrange multiplier (mass conserv.)

ε: boundary width of the cell (interface)

β: coupling param. (e.g., adhesion

strength, growth rate of filaments)

Due to active terms Pε · ∇ρε, β∇ρε,no gradient flow structure

ρ=1

ρ=0

Sharp Inter face

1=ρSharp Inter face

=0ρΩ

ɛ

Appendix: Meaning of each term


Mathematical novelty

Specific mathematical features of PF cell-motility model (1)-(2)

(a) gradient coupling terms (active): Pε · ∇ρε in (1), β∇ρε in (2);

(b) mass conservation constraints (Sternberg, Rubenstein for A-C)

led to both novel results and analysis techniques.

(a) and (b) prevent use of classical techniques:

I maximum (comparison) principle leading to viscosity solutiontechniques (Evans, Barles, Soner, Souganidis).

I Γ-convergence of grad. flow, G-L energy (Serfaty/Sandier).

Our system: no comparison principle, no gradient flow structure.

Gradient coupling leads to new terms in the curvature motion Eq.:

Vn = κ−1

|Γ|

ˆκds 7→ Vn = κ+

β

c0Φ(Vn)−

1

|Γ|

ˆΓ

(κ+

β

c0Φ(Vn)

),

where Φ(V ) is known nonlinear function due to active terms.


1st result: equation for sharp interface (ε→ 0)

Theorem 1 (Sharp interface eqn via formal asymptotics as ε→ 0)

Sharp interface limit ε→ 0 of (1)-(2) leads to evolution of interface Γ(t):

Vn = κ+β

c0Φ(Vn)−

1

|Γ(t)|

ˆΓ(t)

(κ+

β

c0Φ(Vn)

), (3)

Vn: inward normal velocity of Γ(t); κ: curvature of Γ(t);c0 absolute constant determined by the double-well potential W ;Φ(V ) is given by:

Φ(V ) =

ˆ(Ψ0 · n) (θ′0(z))

2dz , (4)

Ψ0(z;V ) solves −Ψ′′0 (z)− VΨ′0(z) + Ψ0(z) = −nθ′0(z) with Ψ0 → 0 as |z| → ∞.

θ0(z): standing wavesolution of the Allen-Cahnequation θ′′0 = W

′(θ0)For W (ρ) = 1

4ρ2(1− ρ)2:

θ0(y) =12(tanh y

2√

2+ 1) y

O

Φ

V

(V)


Solvability & critical βcrit

Nonlinear, nonlocal SIL eqn for interface from phase-field (1)-(2):

V =β

c0Φ(V ) + κ− λ, where λ(t) := 1

|Γ(t)|

ˆΓ(t)

(κ+

β

c0Φ(V )

)(5)

Define F (V ) := V − βc0 Φ(V ); SIL equation: F (V ) = κ− λ︸︷︷︸independent of V

Solving (5) for V : βcrit = sup {β′ : F (V ) is monotone for all β < β′}β ≤ βcrit: uniqueness; β > βcrit: non-uniqueness. (κ,

´Γ

indep. of V )

VV

Subcritical: ββcrit

Monotone InvertibleUnique solution

Non-monotone Non-invertibleNon-unique solution

V- Φ(V)βc0V- Φ(V) βc0

β subcritical: contraction principle ⇒ uniqueness and smoothness of Vn• 2D case: soln existence, TW non-existence (M. Mizuhara et al., Physica D)

β supercritical: rise of asymmetry, non-uniqueness and hysteresis in 1DL. Berlyand Hierarchy of PDE Models of Cell Motility

2nd result: 1D justification of sharp interface limit

Introduce a 1D analogue of the 2D model:∂ρε∂t

= ∂2xρε −W ′(ρε)

ε2− Pε∂xρε +

F (t)

ε, x ∈ R1

∂Pε∂t

= ε∂2xPε −1

εPε − β∂xρε.

(6)

(7)

F (t) models volume preservation & curvature effectsTwo term representation of solution ρε (not an asymptotic expansion)

ρε(x , t) = θ0

(x − xε(t)

ε

)+ εψε

(x − xε(t)

ε, t

)+ εuε

(x − xε(t)

ε, t

), (8)

ψε explicitly constructed; uε and xε new unknowns. xε chosen in ε-interface sothat

´θ′0(y)uε(y , t)dy = 0 (that allows for Poincaré inequality for uε).

”Well-prepared” IC: ρε(0, x) = θ0(xε

)+ εvε

(xε

), Pε(0, x) = pε

(xε

)(9)

with ‖vε‖L2(R) + ε‖vε‖L∞(R) + ‖pε‖H1(R) < C . (localization of Pε) (10)Proposition 2 (justification of the ansatz (8))

Let ρε,Pε be a solution of (6)-(7) with ”well-prepared” IC (9)-(10). Thenthere exists ε-interface xε(t) ∈ C 1 s.t. (8) holds with ‖uε(·, t)‖L2(R) < c.(xε(t) is described in the proof, see reduced system (11)-(12))


Passing to the sharp interface limit ε→ 0Vε(t) := ẋε(t), plug repr. (8) into (6)-(7) to get intermediate system

c0Vε(t) = −F (t) +ˆ

(θ′0)2Pεdy + o(1) (11)

ε∂Pε∂t

= P ′′ε + VεP′ε − Pε − βθ′0 + o(1), (12)

Passing to the limit ε→ 0 in (12) naively: drop ε∂tPε term to get anonlinear eqn for P0 = limPε which yields the sharp interface eqn forV = limVε (note βΦ(V ) :=

´(θ′0)

2P0dy):

c0V (t) = −F (t) +ˆ

(θ′0)2P0dy (13)

0 = P ′′0 + VP′0 − P0 − βθ′0, (14)

Justification of passing to the limit:

I for small β: via Contraction Mapping Principle;

I for larger β: Linearize, then stability analysis. Brings surprises(non-uniqueness, rise of asymmetry, hysteresis).


Stability of Reduced System (11-12) around SIL (13-14)

1D Sharp Interface Equation: c0V (t)− βΦ(V (t)) = −F (t). (15)

Jump:

Jump:

• stable velocities •unstable velocities

If LHS of (15) is non-monotone in V , then(15) does not define uniquely evolution ofV . Branch determined by initial V (t0) thatsolves c0V (t0)− βΦ(V (t0)) = −F (t0), finda continuous function V (t) which solves(15) on an interval [0,T?] – then: jump andhysteresis loop

(stability for (12))

V (t) ∈ SD- set of stable velocities ([Vmax ,Vmin] - unstable/forbidden):SD := {V : σ(T (V )) ⊂ {Reλ > 0}} , (16)

for linearized operator T (V ): T g := −g ′′ − V0g ′ + g −{´

(θ′0)2gdy

}P ′0,

around steady state (V0,P0) of (11)-(12) which solves:

0 = P ′′0 +V0P′0−P0−βθ′0, c0V0 =

ˆ(θ′0(y))

2P0(y)dy−F = βΦ(V0)−F .


Hysteresis loop in 1D: SIL vs PDE (numerics).

Sharp Interface Equation: V (t)−βΦ(V (t)) = F (t), 0 ≤ t ≤ 1. (17)Choose F↓(t) = Fmax + (Fmin − Fmax)t (decreasing, red),F↑(t) = Fmin + (Fmax − Fmin)t (increasing, blue);Fmin = −2.25, Fmax = −1.0; F↓(t) = F↑(1− t)Arrows show in what direction the system (V (t),F (t)) evolves.

Hysteresis: Evolution of the interface depends on the history!Cf. jump in velocity − sub-critical transition in Ziebert et al. 2012


3rd result: Traveling waves (TW) in 1D Model

Goal: Find Traveling Waves − solutions of (6)-(7) of the formρε(x , t) = ρε(x − Vt), Pε(x , t) = Pε(x − Vt). (18)

Proposition. (standing wave solutions for any β)∃ localized standing waves (V = 0) for any β, ε ≤ ε0 of “Π shape”Heuristics: due to symmetry, front and back are trying to move in opposite

directions with the same velocities, thus, cell does not move. In contrast:

Theorem 3 (Existence of TW with finite V for only large β)

Let W (ρ) and β be such that

2c0V = β(Φ(V )− Φ(−V )) (19)has a root V = V0 > 0 and 2c0 6= β(Φ′(V0) + Φ′(−V0)) (IFT). For suff.small ε > 0, ∃ localized TW with V = Vε 6= 0; Vε → V0 6= 0 as ε→ 0.

Bifurcation in β: for β > βcrit a non-trivial TW appears. Thm guarantees ∃ ofTW: observable steady motion without external stimuli.Remark 1. In Theorem, it is crucial that (19) has a non-zero solution V0:• not true for symmetric potential W (ρ) = ρ2(ρ− 1)2;• does hold for asymmetric potentials, e.g., W (ρ) = ρ2(ρ− 1)2(1 + ρ2).


Supercritical vs Subcritical, drastic difference

Subcritical (β ≤ βcr): M. Mizuhara et al., Physica D (2016)

I Cell evolves to a stable circular shape (as in mean curv. motion) buthas shift due to non-linear term;

I No traveling waves.

Supercritical (β > βcr):

I Instability of zero velocity ⇒ circular shape (V = 0⇒ κ = const.)is no longer stable: “system prefers asymmetry/motion,” i.e.,spontaneous breaking of symmetry − circle is no longer stable;

I Bifurcation from standing to traveling wave solutions;

I Hysteresis loop in 1D.


Part II:Free Boundary Problems (FBP) of Cell Motility:

work in progress

The FBP arising in modeling of cell motility differs from bothclassical Stefan like problems and more recent FBP of tumorgrowth. It is nonlinear, boundary motion is curvature dependent,and the TW solutions have constant shape and size whereas intumor growth problems, the domain size is changing. Thesefeatures present a considerable challenge and were addressednumerically in previous works1.Our goal is to develop an analytical approach to this problem.Specifically, find the standing wave solutions by analysis of aLiouville type PDE with exponential nonlinearity, and find travelingwaves by applying topological consideration (Leray-Schauderdegree). Results for both PF and FBP models will be compared.

1e.g., Barnhart, Lee, Allen, Theriot, Mogilner (2015)L. Berlyand Hierarchy of PDE Models of Cell Motility

Free Boundary Problem

Cell occupies moving domain Ω(t), boundary Γ(t) evolving with normal veloc.:

V⊥ = ν · u + λp − 2τ0κ on Γ(t), λp =1

|Γ(t)|

ˆΓ(t)

2τ0κ− ν · u (20)

with actin flow veloc. u, curvature κ (membrane bending)λp - mathematically: area preservation; physically: actin polymerization(20) is coupled to PDEs for u and myosin density m:(µs

3+ µb

)∇ div u + µs∆u + k∇m = ζu in Ω(t), σν = 0 on Γ(t) (21)

∂m

∂t= Dm∆m − div (um) in Ω(t), Dm

∂m

∂ν= (u · ν − V⊥)m on Γ(t) (22)

(21): “Elasticity eqn”, stress, (kmI − active stress)σ = ((µb − 23µs)divu + km)I + µs(∇u +∇u

T )force balance divσ = ζu (23)

(22): Advection-diffusion with no-flux BC (in moving frame)• Unknowns: Ω (or equiv.: Γ), u, m• µb, µs : bulk- and shear viscosity, k: myosin contractility,ζ: adhesion, Dm: diff. coeff. of myosin, τ0: membrane rigidity,|Ω|: (fixed) cell size• Contrast classical Stefan problem (SP): (20)-(22) nonlinear,V⊥ depends on curvature (absent in SP), area preservation

muV

Ω(t)

Γ(t)

T


Vanishing shear viscosity and traveling wave (TW) problem

Experimentally: µs � µb ⇒ vanishing shear viscosity µs = 0. Instead of u in(21) get eqn for scalar stress σ using (23): σ = σI = (µbdivu + km)I.Force balance divσI = ζu becomes Darcy’s Law ∇σ = ζu

0 = ∆σ − σ + αm in Ω(t), σ = σν = 0 on Γ(t) (24)∂tm = ∆m −∇·(m∇σ) in Ω(t), ν ·(∇m −m∇σ) = 0 on Γ(t) (25)

V⊥ = ν ·∇σ − βκ+ λp on Γ(t), λp = −

´Γ(t)

(ν ·∇σ − βκ)|Γ(t)| (26)

Fixed parameters: α, β, other constants rescaled to 1.

Remark: (24)-(25) is Keller-Segel type system due to cross-diffusion ∇·(m∇σ).TW ansatz: Ω(t) = Ω0 + ~V t, u(x , t) = u(ξ), m(x , t) = m(ξ), ξ = x − ~V t (27)

TW problem for σ(ξ),m(ξ) in unknown domain Ω0 (position, shape fixed):

0 = ∆σ − σ + αm in Ω0, σ = 0 on Γ0 = ∂Ω0 (28)

0 = ∆m −∇·(m(∇σ − ~V )) in Ω0, ν ·(∇m −m(∇σ − ~V )) = 0 on Γ0 (29)

ν · ~V = ν ·∇σ − βκ+ λp on Γ0, λp = −´

Γ0(ν ·∇σ − βκ)|Γ0|

. (30)

integrate (29) to m(ξ) = m0eσ−~V·ξ, plug this into (28), choose wlog. ~V = V e1

(29) 0-flux BC⇒ unknown normalization m0 for myosin mass M (IC for (22)).L. Berlyand Hierarchy of PDE Models of Cell Motility

Reduced TW problem to Liouville type PDE and SW

0= −∆σ + σ − Λeσ−Vξ1 in Ω0 σ = 0 on Γ0 = ∂Ω0 (31)

with kinematic cond. in moving frame V ν1 = ν ·∇σ − 2βκ+ λP on Γ0 (32)Unknowns are σ, Ω0, V , and Λ := αm0 (m0: const. of integration); (32) is (30).Non-linear E.V. problem for Λ when V = 0 (SW), Ω0 = DR a disk of radius R

Proposition (Existence and analyticity of SW for (20)-(22))

Let Ω0 = DR , V = 0 in (31). Then, there exists a curve (family) of SW solns(σs ,Λs) emanating from σ0 ≡ 0, Λ0 = 0 with analytical dependence on s fors ∈ (0, s∗(R)). Local uniqueness: in a neighborhood of curve, no other SWsolns (at s∗ existence or uniqueness breaks; simulations s∗ =∞, red curve).

Remark: If Ω0 = DR and V = 0, (32) is satisfied due toradially symmetric solns. Thus u = ∇σs(|x |), m = Λseσs (|x|)are a SW solution of (20)-(22) - solvability of SW problem.Idea of pf: Linearized operator for (31) with V = 0:

−∆w + w − Λseσsw in DR w = 0 on ∂DR . (33)0 can be an E.V. (i.e., IFT fails: crit. point).Crandall-Rabinowitz⇒ soln. curve extends beyond first crit.point (i.e., IFT conclusions hold: existence & uniqueness).

Appendix: further details Λ

||σ||

Λ

(σ ,Λ )

s* s*(σ ,Λ )

s* s*


Non-radial SW solutions

For the problem (20)-(22) a family of non-radial SW solutions bifurcatesfrom a family of radial ones in Ω = DR (β is stiffness of membrane inkinematic BC: V ν1 = ν ·∇σ − βκ+ λP)Theorem 2.1

Given both R > 0 and ` = 2, 3, . . . , for sufficiently small β > 0 there is afamily of steady states solutions of (20)-(22) whose domain Ω is given by

Ω = {x = (R + ρδ(ϕ)) cosϕ, y = (R + ρδ(ϕ)) sinϕ} (34)with ρδ = δ cos `ϕ+ o(δ), δ > 0: small param. (soln. family in δ).

Non-radial steady states: “flowers” indexed by # of “leaves” ` (below ` = 2 & ` = 8)

Appendix: more discussion on non-radial SW


Discussion of the theorem

For a given ` = 2, 3, ... bifurcation occurs along e.functionscorresponding to linearization around radially symmetric solns. Dueto radial symmetry e.f. are cos `ϕ, ` 6= 1 to exclude infinitesimaltranslations (trivial motion due to translation of coordinatesystem). For fixed ` = 2, 3, ... bifurcation occurs only for suff.small β ≤ β0(`). Thus, for a given β Theorem guaranteesexistence of only finite number of non-radial SW solns (indexed byfinite number of `). Proof relies on Leray-Schauder degree theory(C-R also works).

Appendix: more on finite # of non-radial SW


Main result: Bifurcation of TW from SW

Theorem (Bifurcation from standing waves to TW)

Traveling wave solutions of the FBP (20) - (22) bifurcate from the SWsolutions at a bifurcation value sb(R) < s

∗(R).

Theorem provides: • existence of TW that originate from SW bycontinuous change of physical parameters (myosin density).

• General description of physical parameter regimes where TW+SW existor only SW exist (extending results of Truskinovsky in 1D to 2D); notrivial SW as in 1D Tr.

• Explains numerical results in 2D for steady motion at low adhesion (ourfurther results show existence of non-radial SW in 2D not observednumerically for full PDE system).

• Explains switching from SW at high adhesion, ζ, to TW at lowadhesion: parameter s is determined by total myosin massM = Λs

´DR

eσs . In particular, to sb corresponds crit. myosin mass Mb atwhich the bifurcation occurs. The value Mb depends on parameter ζ.


Idea of Proof

1. Solve Liouville Eqn. (31) with σ = 0 on ∂Ωρ(Ωρ = {(r , φ) | r < R + ρ(φ)}) by perturbation expansion in small ρ andV (ρ = 0, V = 0 − radial solution). Existence by IFT and C-R, remainsto satisfy (32).

2. Try to satisfy kinematic BC (32) with this perturbation expansion(linearization) leading to

2ΛsπR2 − Λs

ˆDR

eσsdξ −ˆDR

σsdξ = 0 (35)

sb solves (35) - necessary condition for bifurcation point.

3. To establish sufficiency use exact nonlinear kinematic condition (32).Existence of TW solutions by Leray-Schauder degree Appendix: Leray-Schauder :X = F (X ; s), X = (V , ρ) with compact mapping F .

4. Show sb < s∗. For small R, based on results of T. Suzuki, C. Bandlefor ∆σ + Λeσ = 0. For large R, based on uniform bounds on maximalΛ = Λ̄ > 1/e (from (31) with V = 0).


Example: asymmetric shape of emergent TW

Compute first three terms for expansion in small V of soln of FBP ofLiouville type eqn (31) for R = 4 when bifurcation occurs from minimalsolns: σ(ρ, ϕ,V ) = V 0σ0(ρ, ϕ) + Vσ1(ρ, ϕ) + V

2σ2(ρ, ϕ), then σ0 = 0In polar coordinates this shape is given by

ρ(φ) = 4− .5 cos 2ϕ− .1 cos 3ϕ (36)

where ϕ ∈ [−π, π) is the polar angle.

- 3 - 2 - 1 1 2 3

- 4

- 2

2

4

- 4 - 2 2 4

- 4

- 2

2

4

Bifurcation

Figure: Radially symmetric SW (left) bifurcates to non-radial TW (right)


Summary: Analytic Study of Cell Motility PDE Models

Bio questions: (i) persistent motion & (ii) breaking of symmetry

(A) Phase field (PF) system with active terms

1. Novel non-linear, non-local sharp interface equation derived

2. Critical values of parameters (adhesion/protrusion) predictedaddressing bio issues;

3. Existence of TW for β > βcrit in 1D; TW model steady motion;

4. Standard perturbation does not work due to active terms, developed“weak” approach: implicit tracking of front (cf. homogenization)

(B) Keller-Segel type system with nonlinear/nonlocal Free BC

1. (i), (ii) addressed: existence of non-radial SW and bifurcation fromradial SW to TW; breaking of symmetry captured analytically -ansatz for TW with small V

2. Techniques: Topological Leray-Schauder argument applied toLiouville-type eqn with extra term


Current and future work

I Free boundary model with non-zero shear viscosity µs ;vectorial structure of problem becomes important.

I Stability of standing and traveling waves.

I Adding more physical/biological features to the model.

I Numerical study of global bifurcation picture.


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