Modeling through nonlinear flux limited spreadingjcarrill/Cuba/Lectures/Sanchez.pdf · Modeling...

Modeling through nonlinear flux limited spreading

O. Snchezjoint works with J. Calvo, J. Campos, V. Caselles, P. Guerrero, J.

Soler, A. Ruiz i Altaba and I. Guerrero Labs.

Dept. de Matemtica AplicadaUniversidad de Granada.

Mathematical modeling in Biology and MedicineSantiago de Cuba, 8-17 July 2016

O. Snchez et al. (UGR) Nonlinear flux limited spreading Sant. de Cuba, 8-17 July 2016 1 / 26

Biological motivation

D-V neural patterning

Dorsoventral spinal cord patterning of the chick embryo

(Gilbert)

In Drosophila, Hh plays the same role than Shh in Vertebrates.



Models of signal propagation and transduction

How do morphogen gradients form and propagate?

Diffusion equation (Ficks law) is frequently used to describemorphogen propagation and formation of concentration gradient

ut = xu

(Fick (1855), Brown (1828), Einstein (1905),...)

randomnesssmall particles

How is the signal interpreted by the responding cells?Law of mass action to describe rates of change in proteinconcentrations and gene codifications.

Reaction-diffusion equations

ut = xu + f (u(t , x), ...)

(Turing, 1953, Meinhard, Wolpert, 1969, Lander, 2002, ...)



Models of signal propagation and transduction

How do morphogen gradients form and propagate?

Diffusion equation (Ficks law) is frequently used to describemorphogen propagation and formation of concentration gradient

How is the signal interpreted by the responding cells?Law of mass action to describe rates of change in proteinconcentrations and gene codifications.

Reaction-diffusion equations

ut = xu + f (u(t , x), ...)

(Turing, 1953, Meinhard, Wolpert, 1969, Lander, 2002, ...)



Modeling Shh by linear diffusion

(Saha and Schaffer, Development, 2006)

[Shh]t

= x [Shh]

+ koff [Ptc1Shhmem] kon[Shh][Ptc1mem](t , x)

Shh Ptc1 Gli1

Ptc1cytGliAct

GliRepSmo gli1

Gli1Act, Gli3Act

Ptc1ptc1

Ptc1mem

Gli3Rep





[Shh]t

= x [Shh]


(Loading Video...)


ShhEvolution.mpgMedia File (video/mpeg)




[Shh]t

= x [Shh]


Drawbacks of the modelTransport modeled with diffusion equation: unphysical spreadingout of morphogen to all the neural tube soon after secretion.

(H.C. Park, J. Shin, B. Appel, Development, 2004)

The concentration of Shh received by the cells and the time ofexposure are of similar relevance. (J. Briscoe et al., Nature, 2007)


Transport modeling

Alternative description of the transport mechanism

Substituting Ficks law by the Cattaneo law gives

utt + ut = u

In 1992, M.B. Rubin showed that Cattaneos model of hyperbolicheat conduction violates the second law of thermodynamics.

Changing the classical diffusion term by a power law diffusion ofporous medium type

ut = .(umu)(Murray, 2002, Vzquez 2007)

Ph. Rosenau (1992), from the observation that the speed of soundis the highest admissible free velocity in a medium, derived

ut = .

|u|u|u|2 + 2c2 |u|

2

c

u (RHE)


Transport modeling



utt + ut = u




ut = .

|u|u|u|2 + 2c2 |u|

2

c

u (RHE)


Transport modeling



utt + ut = u



Ph. Rosenau (1992), from the observation that the speed ofsound is the highest admissible free velocity in a medium, derived

ut = . (Vu) s.t. |V | c

ut = .

|u|u|u|2 + 2c2 |u|

2

c

u (RHE)


Transport modeling



utt + ut = u




ut = .

|u|u|u|2 + 2c2 |u|

2

c

u (RHE)


Transport modeling

Alternative description: flux limited operators

Different operators can be deduced by considering optimal masstransportation, kinetic derivations, porous versions, ...

1Dexamples:

ut =

|u|mux|u|2 + 2c2 |ux |

2

x

m 1 (PRHE)

ut =

|u|(um)x1 + 2c2 |(um)x |

2

x

m 1 (FLPME)

The mathematical properties of this equation and related models havebeen analyzed thoroughly. Well-posedness (especially uniqueness) isproperly dealt with in the framework of entropy solutions (Andreu, Caselles,Mazn, Moll, Calvo,... See references in Calvo et al., EMS Surv. Math. Sci, 2015)


Transport modeling

Qualitative behavior of flux limited solutions

Regularity

Model RHE FLPMESupport growth c cDiscontinuities [0,) [0,T [

Smothness ( u > 0) [0,) [0,T [150 J. Calvo, J. Campos, V. Caselles, O. Snchez and J. Soler

A)u.t; x/

x

1

2:51 4

B)u.t; x/

x

1

42:51

Figure 1. Numerical evolution of a stepwise initial condition by the relativistic equation (1.5) inA), and the flux-saturated porous media equation (2.9) with m D 2 in B). In both examples wehave taken D c D 1 and t 2 0; 0:9 (the smaller the height, the more advanced the times). Thetime step between dierent profiles is 0:05 for the three first profiles and 0:2 for the rest in order tocapture the dierent velocities. Note the instantaneous regularization at the interior jumps althoughthis is not the case at the boundary. We remark the persistence of the discontinuous jumps at theboundary in A), which disappear in B). It also interesting to observe that velocity of support isconstant in A) against the fact that it decreases in B).

In fact, we will display a number of simulations not just here but rather throughout thetext. We focus in this section (Figures 1, 2 and 3) in both pure flux-saturated equationsand some porous media variants; later on we will show some simulations where couplingwith a reaction term or even with a system of ODEs is featured. Regardless of thesecharacteristics, the set-up used for those numerical simulations that are presented in thedocument is always the same. The numerical solution considers a spatial discretizationof the flux-saturated transport equation by using a fifth-order finite dierence WENO(Weighted Essentially Non-Oscillatory) scheme [92] with LaxFriedrichs flux splitting[84]. For the time evolution we use a fourth order RungeKutta method, which alsoallows to deal with possible delay phenomena. A spatial grid between 1000 and 2000points with an appropriate CFL condition is considered.

The previous procedure is but one among a number of dierent possibilities. There arein the literature dierent numerical approaches to flux-saturated equations, among whichwe mention [20, 129, 52, 50, 57, 67, 68, 102, 122].

4. Mathematical preliminaries: The bounded variation scenario

In this section we introduce a number of tools that are needed to set up the well-posednessframework in [11, 12] and some of its extensions. As pointed out in the introduction and inthe previous numerical examples, in general we may not expect solutions of flux-saturatedequations to be more regular than u 2 BV..0; T Rd /. This makes operations like in-tegration by parts already involved. But in fact this may be even worse. First, there isno particular reason why ut should even be a Radon measure, which creates a number of

(Caselles, Carrillo, Mol, PLMSoc, 2013, Calvo et al. to appear JEMSoc)


Transport modeling

Qualitative behavior of flux limited solutions

Waiting time

Model PRHE FLPMESupport growth cmu0m1 c

Waiting time m > 1 m > 1Smothness !!! !!!

Flux-saturated porous media equations 151

A)u.t; x/

x

1

2:51 4

B)u.t; x/

x

1

42:51

Figure 2. Numerical evolution of an initial condition given by a continuous polynomial spline.Plot A) depicts the evolution by (2.11), while plot B) shows the evolution by (2.9). In both casesm D 2 and D c D 1. The time step between successive profiles is 0:48. In both cases we finda waiting time for support spreading which is longer for the case of (2.9). Note that the cusp isregularized in both cases but a discontinuity occurs in the derivative of the solution when slidingthrough the initial profile. The result is a continuous profile for (2.9), which reduces its velocity,and the emergence of a jump discontinuity in the case of (2.11), which is a moving front in whichthe velocity of propagation depend on the parameters of the system via a RankineHugoniot-typecondition. Therefore, in both cases there is a smoothing process but, simultaneously, another oneof singularization emerges either in the derivative in B) or as a jump in A).

technical issues (we will treat this in more detail in Section 10.2). And second, the de-generacy of the equation may spoil the previous spatial regularity on the zeroth level set.Then it will be mandatory to avoid this set when dealing with certain delicate technicalissues related with well-posedness. This motivates the introduction of a specific set oftruncation functions that will be essential in order to construct the functional frameworkin which well-posedness can be proved. Another set of specific truncation functions willbe required for the sole purpose of showing uniqueness, as the extremely low regularity ofsolutions requires to use Kruzkovs doubling variables methodology. This proof uses verycomplicated combinations of terms involving functions with extremely low regularity. Avery specific functional calculus needs to be defined in order to make sense of the previ-ous. In particular, lower semicontinuity results for energy functionals in this degenerateframework will be needed.

The required toolkit to cope with the above technical issues will be introduced in thissection. Its contents are extracted from [11, 12, 17, 13, 56, 62].

Let us take the opportunity to state here some generic purpose notations. Throughoutthe document B.x; r/ denotes an open ball centered at x with radius r . Let us denoteby Ld and Hd1 the d -dimensional Lebesgue measure and the .d 1/-dimensionalHausdor measure in Rd respectively. Given an open set in Rd we denote by D./the space of infinitely dierentiable functions with compact support in . The space ofcontinuous functions in will be denoted by C./ (resp. Cc./ for continuous functionswith compact support in ). Likewise, Ck; denotes the class of k-times dierentiablefunctions whose kth derivatives are Hlder-continuous with exponent . Lebesgue and

(Calvo et al. to appear JEMSoc)


Traveling waves

Traveling waves in reaction-diffusion models

At the continuous level, diffusion equations coupled to reaction termsare able to display wave-like phenomena.

For instance, the FKPP model

ut = uxx + ku(1 u).

displays classical traveling waves u(t , x) = u(x t) for wavespeeds 2

k. (Fisher; Kolmogoroff, Petrovsky, Piscounoff, 1937)

13.2 FisherKolmogoroff Equation 441

U + cU + U(1 U) = 0, (13.8)

where primes denote differentiation with respect to z. A typical wavefront solution iswhere U at one end, say, as z , is at one steady state and as z it is at theother. So here we have an eigenvalue problem to determine the value, or values, of csuch that a nonnegative solution U of (13.8) exists which satisfies

limz U(z) = 0, limz U(z) = 1. (13.9)

At this stage we do not address the problem of how such a travelling wave solutionmight evolve from the partial differential equation (13.6) with given initial conditionsu(x, 0); we come back to this point later.

We study (13.8) for U in the (U, V ) phase plane where

U = V, V = cV U(1 U), (13.10)

which gives the phase plane trajectories as solutions of

dVdU

= cV U(1 U)V

. (13.11)

This has two singular points for (U, V ), namely, (0, 0) and (1, 0): these are the steadystates of course. A linear stability analysis (see Appendix A) shows that the eigenvalues for the singular points are

(0, 0) : =12

[c (c2 4)1/2

]

{stable node if c2 > 4stable spiral if c2 < 4

(1, 0) : =12

[c (c2 + 4)1/2

] saddle point.

(13.12)

Figure 13.1(a) illustrates the phase plane trajectories.If c cmin = 2 we see from (13.12) that the origin is a stable node, the case when

c = cmin giving a degenerate node. If c2 < 4 it is a stable spiral; that is, in the vicinity

Figure 13.1. (a) Phase plane trajectories for equation (13.8) for the travelling wavefront solution: here c2 >4. (b) Travelling wavefront solution for the FisherKolmogoroff equation (13.6): the wave velocity c 2.

(Murray, Mathematical Biology)


Traveling waves

Are flux-limited models able to reproduce patterns?

We expect to get front-like traveling waves when we couple such anon-linear diffusion (m 1) to a FisherKolmogorov reaction term:

ut = v0

(u/v0)mux|u|2 + 2c2 |ux |2

x

+ ku(1 u/v0)

Several other models are embodied as limiting cases.


Traveling waves

Catalog of solutions, case m = 1

t

v

t

v

(Campos, Guerrero, Snchez, Soler, Ann. IHP 2013)


Traveling waves

Catalog of solutions, case m > 1

A)u

1B)

u

1

C)u

1D)

u

1

Figure 1: Travelling wave profiles for 2: A) > smooth, B) = smooth, C) (ent,smooth), D) = ent. Vertical dotted lines show singular tangentpoints. These profiles are in correspondence with the orbits depicted in Fig. 2B).

thus they encode processes in which the propagation of information (whateverit may be) takes place at finite speeds see discussions elsewhere [].

The natural context where studying discontinuous solutions to fluxlimitedporous media equations is the BV theory as it has been analyzed in [7, 17, 18]which provides a suitable functional L1-framework that allows to treat suchsingular objects by means of the concept of entropy solutions.

2 Entropy solutions

Our purpose in this section is to state existence and uniqueness of entropy solu-tions of the reaction-diffusion equation (2). Moreover, we also give a geometricinterpretation of the entropy conditions on the jump set of the solutions. Equa-tion (2) belongs to the more general class of flux limited diffusion equations forwhich the correct concept of solution, permitting to prove existence and unique-ness results, is the notion of entropy solution [4, 17]. This class of equationswith F = 0 has been studied in a series of papers [3, 4, 6, 17, 18, 7] and for theso-called relativistic heat equation (m = 1 in (2)) with a FisherKolmogorovtype reaction term in [5]. The notion of entropy solution is described in terms ofa set of inequalities of Kruzhkovs type [31]. As proved in [18] when F = 0, theycan be characterized more geometrically on the jump set of the solution by say-ing that the graph of the function is vertical at those points. Our purpose is toextend these results to the case of (2) and a reaction term of FisherKolmogorovtype. This will be fundamental to construct discontinuous traveling waves thatare entropy solutions of (2).

Our first purpose is to give a brief review of the concept of entropy solution

5

(Calvo, Campos, Caselles, Snchez, Soler, to appear Invent. Math.)


Traveling waves

The traveling wave ansatz

We want solutions connecting the constant states one and zero havinga constant shape:

u(t , x) = u() = u(x t), > 0.

The profiles we are seeking are non-negative functions u() definedon ], [ for some ],+], such that

lim

u() = 1 and u() < 0.

Remark: = and lim u() = 0 are posteriorly deduced.


Traveling waves

Existence FKPP: reduction to a planar system

The traveling profile must solve the following equation:

(u)

+ u + ku(1 u) = 0.

Settingr() = u() ,

the second order ODE is equivalent to a first order planar dynamicalsystem:

u = 1

r ,

r =

r + ku(1 u).

Furthermore, lim u() = 1 and r > 0 by hypothesis.


Traveling waves

Existence: linear analysis around equilibria points

There exist only two equilibriapoints,

(1,0), saddle node.

Eigenvalues: =

2+4k2

Eigenvectors: ( 1 ,1)

(0,0), (im)-proper iff 2

k

Eigenvalues: =

24k2

r

u1

The condition lim u() = 1 implies that the traveling wave profileis determined by the unstable variable.


Traveling waves

Existence: positive invariant region

The existence of an heteroclinic so-lution joining the fixed points is con-sequence of the existence of a pos-itive invariant region for the planarflux.

r

u1

(1, /2)


Traveling waves

Case m>1, Orbits for the singular dynamical system

A) r

u1

1u()

r()

B) r

u1

1

u(smooth)

u(ent)

Figure 2: A) Normalized direction field of the flux related to (29) for = c = 1,m = 2, f(u) = u(1 u) and = 4/3. Dierent types of arrows were usedto stress the fact that the actual flux is singular at the boundaries r = 1 andu = 0. B) Numerical solutions to type I (solid), II (dashed) and III (dotted)orbits of (29) for several values of . The lowest type II and III orbits are thosecorresponding to smooth. The uppermost type II orbit corresponds to ent.The intermediate type II orbit and the uppermost type III orbit correspond toa value 2 (ent,smooth) and are related by the jump law (25).

20


Traveling waves

Characterizing the entropy solution

Distributional solutions to our model having a jump discontinuity mustsatisfy the Rankine-Hugoniot jump condition:

v =F(u)+ F(u)

u+ u

v velocity at which the jump discontinuity movesu values of the solution at both sides of the discontinuityF(u) values of the flux at both sides of the discontinuity


Traveling waves

Characterizing the entropy solution

Distributional solutions to our model having a jump discontinuity mustsatisfy the Rankine-Hugoniot jump condition:

v =F(u)+ F(u)

u+ u

In our particular situation, this reduces to

v = c(u+)m (u)m

u+ u

Entopic solutions must satisfyRankine-Hugoniot jump conditionThey must have infinite slopes at both sides of the possiblediscontinuities (except if u+ = 0)


Traveling waves

Singular traveling waves viewed as attractors

30 Calvo, Campos, Caselles, Sanchez & Soler

The numerical time dependent solutions given in Figure 4A) has the samepower law behavior near the front than the corresponding traveling wave(3.2).

In Figure 4B) the numerical calculations show spontaneous singulariza-tion of solutions and the convergence of an initial data towards a travelingwave solution to the type described in Figure 1C).

A)

u(t, x)

x

1

B)

u(t, x)

x

1

Fig. 4 Dotted lines represent time dependent solutions for (1.2) (the smaller thedots, the more advanced the times). Dashed lines describe traveling wave pro-files obtained from (2.2) for dierent values of . In all cases we have used = c = 1, F (u) = u(1 u) and m = 2. A) Time evolution of a compactlysupported initial condition with null Dirichlet conditions. The traveling wave pro-file depicted, corresponding to ent = 0, 437803, constitutes a super-solution forthe time-dependent solution. B) Time evolution of a regular initial condition, withNeumann boundary conditions. The traveling wave profile depicted corresponds to = 0, 57 2]ent,smooth[.

3.2 The Lp-continuity w.r.t. the wave speed

The purpose of this paragraph is to prove the continuity of the travelingprofiles constructed in Proposition 8 with respect to the wave velocity. Inorder to do that it is convenient to choose a privileged normalization for thetraveling profiles, so that we get a family uN () defined in a unique way. Wedo this as follows:

Definition 5 Let uN with 2 [ent,+1[ be the family of traveling wavesolutions constructed in Proposition 8 and enjoying the following additionalproperties:

If > smooth we set uN (0) = u

(smooth) = u+(smooth) = u(smooth), If smooth > ent we set lim!0 uN () = u(). If = ent we set lim"0 uN () = u

+() and uN () = 0, for > 0.

We assume in this Section that F satisfies

there is some p 1 such that lim infu!0 F (u)up = k 2]0,+1]. (3.3)

1 Spontaneous singularization and convergence towards travelingwaves, which constitute super-solutions for the time-dependentproblem.


Going back to biological models

Modeling morphogenetic responses

Transport of the Shh signal: introduction of flux limiter

[Shh]t

= x[Shh]x [Shh]

[Shh]2 + 2c2 (x [Shh])2


(M. Verbeni, et al. Physics of Life Reviews, 2013)

Shh Ptc1 Gli1

Ptc1cytGliAct

GliRepSmo gli1

Gli1Act, Gli3Act

Ptc1ptc1

Ptc1mem

Gli3Rep




Transport of the Shh signal: introduction of flux limiter

[Shh]t

= x[Shh]x [Shh]

[Shh]2 + 2c2 (x [Shh])2


Shh Ptc1 Gli1

Ptc1cytGliAct

GliRepSmo gli1

Gli1Act, Gli3Act

Ptc1ptc1

Ptc1mem

Gli3Rep




[Ptc1Shhmem ]

t= kon [Shh][Ptc1mem ] koff [Ptc1Shhmem ] + kCout [Ptc1Shhcyt ] kCin [Ptc1Shhmem ]

[Ptc1Shhcyt ]

t= kCin [Ptc1Shhmem ] kCout [Ptc1Shhcyt ] kCdeg [Ptc1Shhcyt ].

[Ptc1mem ]

t= koff [Ptc1Shhmem ] kon [Shh][Ptc1mem ] + kcyt [Ptc1cyt ],

[Ptc1cyt ]

t= kcyt [Ptc1cyt ] + kP Ptr

([Gli1Act ](t ), [Gli3Act ](t), [Gli3Rep(t)]

)Ptc

[Gli1Act ]

t= kdeg [Gli1

Act ] + kGPtr(

[Gli1Act ](t ), [Gli3Act ](t), [Gli3Rep(t)])

Ptc

[Gli3Act ]

t=

g3

1 + RPtc [Gli3Act ]

kg3r1 + RPtc

kdeg [Gli3Act ]

[Gli3Rep ]

t= [Gli3Act ]

kg3r1 + RPtc

kdeg [Gli3Rep ].

where

Ptc =[Ptc10 ]

[Ptc10 ] + [Ptc1mem ], RPtc =

[Ptc1Shhmem ]

[Ptc1mem ]



Numerical experiments

t = 30 hrt = 24 hrt = 18 hrt = 12 hrt = 6 hr

t = 1.5 hr

distance from floor plate (m)

[Shh](n

M)

300250200150100500

35

30

25

20

15

10

5

0


t = 1.5 hr


[Shh](n

M)

300250200150100500

35

30

25

20

15

10

5

0

No instant spreadingReal wave frontsTime to respond



Numerical experiments


t = 1.5 hr


[Gli1A

ct](n

M)

300250200150100500

25

20

15

10

5

0


t = 1.5 hr

distance from floor plate (m)[G

li1A

ct](n

M)

300250200150100500

2.5

2

1.5

1

0.5

0



Numerical experiments: Desensitization

Briscoeexperiments FLS Linear diffusion

2497REVIEWDevelopment 135 (15)

The repressive activities of neural tube GRN proteins might partlyexplain the temporal profiles of neural tube GRN genes. Membersof the network display dynamic changes in their patterns ofexpression during neural tube development (Fig. 2C, Fig. 6). Forindividual genes, this parallels their dependence on different levelsand durations of Shh signaling that is, class I proteins are initiallyexpressed more ventrally in the neural tube but are progressivelyrepressed in a ventral-to-dorsal manner. The order in which class Igenes are repressed is inversely related to their sensitivity to Shhsignaling (Dessaud et al., 2007; Ericson et al., 1997). Thus, Shh

signaling progressively defines the ventral expression boundaries ofclass I proteins. Class II proteins, conversely, are sequentiallyinduced in the ventral neural tube in an order that corresponds totheir increasing requirement for Shh signaling (Dessaud et al., 2007;Jeong and McMahon, 2005; Stamataki et al., 2005). These temporalfeatures of the neural tube GRN are consistent with the importanceof the duration of Shh signaling in ventral neural tube patterning.Together, the data suggest that the spatio-temporal pattern ofexpression for an individual neural tube GRN gene is determined bya combination of uniform positive inputs (e.g. Sox and/or POU

t0 t1

Active Smo

Inactive SmoPtch1

Shh GliAGliA

GliA

High Gli activitor

Medium Gli activitor

Low Gli activitor

GliRGliR

High Gli repressor

Low Gli repressor

Ventral

Dorsal

Nkx6.1+Olig2+GliA

Nkx6.1+

GliA

GliA

Nkx6.1+Olig2+

GliR

Pax7

Pax7+

GliA:GliR

Nkx6.1+GliA

GliA

GliA

Nkx6.1+

Nkx6.1+

GliR

GliR

Pax7+

Pax7+

t2

Nkx6.1+Olig2

Nkx2.2+

Nkx6.1+Olig2+

Nkx6.1+

GliA

GliA

GliA

GliA:GliR

GliR

Pax7

Pax7+

3

2

1

3

2

1

3

2

1

Time

Gli a

cti

vit

y

4

3

2

1

4 4 4

A B

Fig. 6. A temporal adaptation model for interpreting graded Shh signaling. (A) A model for the spatial and temporal specification ofprogenitor cells during exposure to Shh secreted from a ventral source. At early time points (t0), Ptch1 expression levels (brown receptor) in neuralprogenitors are low, consequently low levels of Shh protein (blue) are sufficient to bind the available Ptch1. This produces high levels of Smo signaltransduction (green) and, consequently, high levels of positive Gli activity (GliA, red), even in cells that are exposed to a low concentration of Shh(cell 3, t0). The upregulation of Ptch1 (and possibly other negative regulators of the pathway) by Shh signaling (t1) increases the level of Ptch1 inresponding cells. As a result, the concentration of Shh necessary to sustain high levels of signal transduction increases with time (t1). In cellsexposed to concentrations of Shh insufficient to bind all of the raised level of Ptch1 (cell 3, t1), the level of GliA declines (GliA, orange). This processof cell-autonomous desensitization continues (t2), resulting in distinct temporal profiles of Gli activity in cells arrayed along the DV axis. In addition,the upregulation of ligand-binding inhibitors of Shh signaling, including Ptch1, results in the sequestration of Shh protein in more-ventral regions ofthe neural tube (cell 1). Both the level and the duration of Shh-Gli activity influence the gene expression profile in responding cells. Low levels of Gliactivity, for example produced by the partial inhibition of the generation of GliR activity, are sufficient to repress Pax7 (cell 4, t1). The duration of Shhsignaling is partially responsible for the distinction between Olig2 and Nkx2.2 induction. High levels of Gli activity induce Olig2 expression (cells 1-2,t1). If the levels of signaling are sustained (cell 1, t2), Nkx2.2 is induced and Olig2 is repressed. By contrast, if the levels of signaling in a cell declineprior to this time point, Olig2 expression is consolidated (cell 2, t2). (B) The response of the indicated cells in A to Shh can be represented as afunction of both Gli activity and the duration of Shh exposure (time). The adaptation of cells to ongoing Shh signaling results in differentconcentrations of Shh producing distinct profiles of Gli activity. Hence, temporal adaptation transforms different concentrations of morphogen intocorresponding durations of increased Gli activity. In this view, the induction of each progenitor state requires exposure to a concentration of Shhabove a defined threshold for a distinct period of time.

DEVELOPMENT

A

time (hours)

[Gli1Act](nM

)

302520151050

2.5

2

1.5

1

0.5

0

B

time (hours)302520151050

25

20

15

10

5

0

[Gli1Act](nM)


References

References

M. Verbeni, O.S., E. Mollica, I. Siegl-Cachedenier, A. Carlenton, I. Guerrero, A. Ruiz iAltaba, J. Soler, Morphogenetic action through flux-limited spreading, Physics of LifeReviews 2013.

J. Campos, P. Guerrero, O. S., J. Soler, On the analysis of travelling waves to a nonlinearflux limited reaction-diffusion equation, Ann. Inst. H. Poincar Anal. Non Linaire 2013.

J. Calvo, J. Campos, V. Caselles, O. S., J. Soler, Flux-saturated porous media equationsand applications. EMS Surv. Math. Sci., 2015

J. Calvo, J. Campos, V. Caselles, O. S., J. Soler, Pattern formation in a flux limitedreaction-diffusion equation of porous media type. To appear Invent. Math.

J. Calvo, J. Campos, V. Caselles, O. S., J. Soler, Qualitative behaviour for flux-saturatedmechanisms: traveling waves, waiting time and smoothing effects. To appear JEMS


Biological motivationTransport modelingTraveling wavesGoing back to biological modelsReferences

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