We discuss the formation of graded morphogen profiles in a cell layer by
nonlinear transport phenomena, important for patterning developing organisms.
We focus on a process termed transcytosis, where morphogen transport results
from binding of ligands to receptors on the cell surface, incorporation into
the cell and subsequent externalization. Starting from a microscopic model, we
derive effective transport equations. We show that, in contrast to morphogen
transport by extracellular diffusion, transcytosis leads to robust ligand
profiles which are insensitive to the rate of ligand production

A. Eldar

A. Gierer

A. D. Lander

A. M. Turing

D. A. Lauffenburger

E. V. Entchev

F. Crick

F. Jülicher

K. Kruse

L. Wolpert

M. González-Gaitán

P. Pantazis

T. Bollenbach

English

arXiv

Bollenbach, T.

Kruse, K.

Pantazis, P.

Gonzalez-Gaitan, M.

Jülicher, F.

MPG.PuRe

Robust formation of morphogen gradients

Crossref

Robust Formation of Morphogen Gradients

We discuss the formation of graded morphogen profiles in a cell layer by nonlinear transport phenomena, important for patterning developing organisms. We focus on a process termed transcytosis, where morphogen transport results from the binding of ligands to receptors on the cell surface, incorporation into the cell, and subsequent externalization. Starting from a microscopic model, we derive effective transport equations. We show that, in contrast to morphogen transport by extracellular diffusion, transcytosis leads to robust ligand profiles which are insensitive to the rate of ligand production

Bollenbach, T

Kruse, K

Pantazis, P

Gonzalez-Gaitan, M

Julicher, F

Spiral - Imperial College Digital Repository

arXiv:q-bio/0412014v1  [q-bio.OT]  8 Dec 2004Robust formation of morphogen gradientsT. Bollenbach,1 K. Kruse,1 P. Pantazis,2 M. González-Gaitán,2 and F. Jülicher11Max-Planck-Institute for Physics of Complex Systems,Nöthnitzer Str. 38, 01187 Dresden, Germany2Max-Planck-Institute for Molecular Cell Biology and Genetics,Pfotenhauer Str. 108, 01307 Dresden, Germany(Dated: June 6, 2018)AbstractWe discuss the formation of graded morphogen profiles in a cell layer by nonlinear transportphenomena, important for patterning developing organisms. We focus on a process termed tran-scytosis, where morphogen transport results from binding of ligands to receptors on the cell surface,incorporation into the cell and subsequent externalization. Starting from a microscopic model, wederive effective transport equations. We show that, in contrast to morphogen transport by extra-cellular diffusion, transcytosis leads to robust ligand profiles which are insensitive to the rate ofligand production.1An essential feature during the development of an organism is the emergence of differentcell types. This process, called cell differentiation, is intimately linked to the position of thecells in the organism. About fifty years ago, Turing suggested the existence of moleculesthat provide positional information for differentiating cells [1]. Turing proposed that thesemorphogens self-organize into patterns which structure the future organism. Pattern for-mation based on reaction diffusion systems has been studied extensively [2, 3, 4]. Today,a number of proteins have been identified to act as morphogens. In contrast to Turing’soriginal suggestion, morphogens are secreted from a spatially localized source and form astable, long-ranged concentration gradient in the adjacent tissue [5]. There, cells detectthe morphogen by receptors located on the cell surface and respond with patterns of geneexpression that depend on the morphogen concentration. In this way, cell fate depends onthe distance from the morphogen source and the interplay of several morphogen gradientsoriginating from spatially distinct sources leads to the complex patterning of developingorganisms. For this to be possible, the gene expression patterns must be formed with highprecision and also need to be robust, i.e. they should not depend sensitively on parame-ters that are likely to fluctuate. Robust gene expression can result from the robustness ofthe morphogen concentration gradient [6, 7] or can be achieved by other mechanisms [8].Interestingly, different species use very similar morphogens and a given species can use thesame morphogen at different stages during development. An example are morphogens ofthe TGF-β super-family which exist in a wide range of organisms ranging from the fruit flyDrosophila to humans.The formation of morphogen gradients is still poorly understood. In particular, it isnot clear, how morphogens are transported in the tissue adjacent to the source. Severalpossibilities have been proposed. Secreted morphogen molecules might simply diffuse in theextracellular space between cells [9] and for some morphogens this indeed seems to be thedominant form of transport [10]. However, there is growing experimental evidence againstsimple diffusion for several morphogens [11, 12]. Other transport mechanisms, that havebeen proposed, include the bucket brigade, where molecules are ”handed over” from one cellto adjacent cells [13], or transport by cell displacements in the tissue [14].Another possibility is suggested by recent experiments on the morphogen Dpp [11]. Dppbelongs to the TGF-β super family and patterns the developing wing discs of Drosophila.The wing disc is a sheet formed from one layer of cells that is the precursor of a wing. Fluo-2rescently labeled Dpp was tracked in wing discs where it is secreted by a narrow strip of cellsfrom where it spreads into the adjacent tissue. If all cells adjacent to the secreting cells aredefective in the internalization of the morphogen into the cell, Dpp gradients extend onlyabout two cells from the source as compared to 30 cells in a non-mutant fly. If only cells ina small patch close to the source show this defect, a pronounced transient depletion of themorphogen is observed behind the mutant patch during gradient formation. This suggeststhat Dpp is not simply diffusing through the extracellular space. It has been suggested thatthese observations can be described by a scenario where Dpp diffuses in the extracellularspace, binds to and is released from receptors [15]. However, an analysis which takes intoaccount more recent experimental results indicates that free diffusion alone cannot accountfor the observations [16]. An alternative suggestion is that long range transport of Dpp isgenerated by transcytosis, i.e. by repeated rounds of morphogen binding to cell surface re-ceptors, internalization into the cell and subsequent externalization and release of the ligandfrom the receptor at a different point on the cell surface [17]. A simple model to describetranscytosis is illustrated in Fig. 1. Ligands spread from a stripe of secreting cells into atwo-dimensional tissue. However, in many situations a one-dimensional description of ligandprofiles as a function of the distance x to the source is appropriate [15, 16]. Transcytosis, canbe characterized by rates kon and koff for receptor binding and unbinding of free ligands, aswell as bint and bext for internalization and externalization of ligand-bound receptors. Inter-nalized morphogens are degraded with rate bdeg. The free diffusion of ligands in the narrowspace between cells is characterized by the diffusion coefficient D0. Diffusion alone does notlead to efficient long-range transport if the diffusion length ξd = (D0/edeg)1/2 is smaller thanone or several cell diameters. Here, edeg is the extracellular degradation rate.On scales large compared to the cell diameter a, this model together with the assumptionthat receptors are also synthesized and degraded in the cells, leads to effective transportequations for the number densities λ(t, x) of morphogens and ρ(t, x) of receptors given by∂tλ = ∂x(D(λ, ρ)∂xλ−Dρ(λ, ρ)∂xρ)− k(λ, ρ)λ (1)∂tρ = νsyn(λ, ρ)− νdeg(λ, ρ)ρ . (2)Here, D(λ, ρ) is an effective diffusion coefficient and k(λ, ρ) an effective degradation ratethat depend on both the ligand and receptor concentrations. Furthermore, a term existswhich describes ligand currents induced by gradients of the receptor concentration. We3assume for simplicity that cells are non-polar such that externalization occurs with equalprobability at any place on the cell surface. As a consequence, the resulting transport isnon-directional. The kinetics of the receptor density is characterized by effective source andsink terms νsyn(λ, ρ) and νdeg(λ, ρ). The dependences of the effective diffusion coefficientD, the degradation rate k, the coefficients Dρ, νsyn and νdeg on the ligand and receptorconcentrations can be calculated explicitly [18]. In order to keep our discussion simple, wefocus here on the case where the surface receptor number per cell R is maintained constant.Then, ligand transport is described by a single nonlinear diffusion equation∂tλ = ∂x(D(λ)∂xλ)− k(λ)λ . (3)Explicit expressions of D(λ) and k(λ) are given in [22]. The coefficients are displayed asfunctions of the ligand concentration λ in Fig. 2. For small λ, D(λ) and k(λ) approachfinite values, and the transport equation becomes linear. For large λ, D(λ) ≈ D0 + c1/λ2with c1 = abintkoffr/(4kon), where r = R/a. The effective diffusion coefficient D(λ) decreasesfor large ligand concentration because receptors are increasingly occupied and not availablefor transport. A maximum of D occurs for intermediate values of λ, if the ligand bindingrate kon exceeds a critical value. This implies that for small λ, D increases for increasingλ. Similarly, the effective degradation rate exhibits for large λ the asymptotic behaviork ≈ edeg + c2/λ where c2 = bdegbintr/bext.We solve Eq. (3) for x > 0, representing the space adjacent to the ligand source, and withboundary conditions j(x = 0) = j0 and j = 0 for large x. Here, j0 is proportional to therate of ligand secretion in the source. In this situation, a graded ligand profile is generatedand maintained in the steady state, see Fig. 3.The effect of the source is captured by the boundary condition on the current j = j0 atx = 0, where j = −D(λ)∂xλ, see Eq. (3). If initially at time t = 0 ligand is absent λ = 0,the ligand spreads for t > 0 into the region x ≥ 0, and builds up a gradient. In the steadystate, the total ligand concentration decreases monotonically with increasing distance fromthe source. The steady state satisfies ∂xj + k(λ)λ = 0 and is reached on a time scale of theorder of 1/k(λ = 0). The corresponding ligand distribution λ(x) obeysx = −∫ λ(x)λ(0)dλ′ D(λ′)/j(λ′) , (4)4where the ligand current j in the steady state is given byj(λ) =[2∫ λ0dλ′ k(λ′)D(λ′)λ′]1/2. (5)For small λ, the ligand profile decays as λ ∝ exp(−x/ξ) with ξ = (D(0)/k(0))1/2. In thecase D0 = 0, the steady state profile has a singularity at x = x∗:λ ∼ (x− x∗)−1(− ln (x− x∗))−1/2 (6)Choosing j0 > 0 ensures x∗ < 0 such that x∗ is not physically accessible. This singularityresults from the asymptotic behavior of the current j for large λ ≫ λT : j(λ)2 ≈ j(λT )2 +2edegc1 ln(λ/λT )+2c1c2(1/λT −1/λ). Here, λT denotes a characteristic ligand concentrationbeyond which the limit of large ligand concentration holds. Therefore the current divergesas j2 ≈ 2c1edeg lnλ. Note, that for edeg = 0 the current reaches for large λ a finite maximalvalue jmax and the corresponding steady state ligand profile diverges as λ ≈ c1/[jmax(x−x∗)].The singular behavior of the steady state profile near x = x∗ has remarkable consequencesfor the robustness of gradient formation. Such a robustness can be quantified by an appro-priate response function of the system [6]. We define the robustness R(j0) := a(j0∂j0x(λ))−1,where the function x(λ) is given by Eq. (4). It becomes large if changes of j0 have littleinfluence on the ligand profile in the steady state. Here, a robustness of R = 1 implies thatunder a 100% increase of j0 the position at which the ligand profile attains any fixed valueis displaced by about one cell diameter a. Thus for R ≥ 1, cells in the target tissue cannotdetect variations of the ligand concentration due to significant changes of j0, while for R ≤ 1the steady state profile is strongly affected, see Fig. 3. Note, that R is independent of λ.Indeed, R can be rewritten asR = a(j0∂j0x)−1 = a∂λ0j0/D(λ0) = ak(λ0)λ0/j0 (7)where λ0 = λ(x = 0) and Eqs. (4) and (5) have been used. The robustness is thus completelydetermined by the ratio of the effective degradation rate and the ligand current at x = 0.High degradation rates and small currents lead to a robust gradient. Using the asymptoticbehavior of the steady state profile for D0 = 0, we find that the robustness increases rapidlyfor large currents j0 as R ∼ j−10 ej20/j2c with j2c = 1/2c1edeg. For small j0, R ≃ a/ξ becomesconstant.5The situation is different if D0 is finite. In this case, both k(λ) and D(λ) are nonvanishingand constant for large λ and consequently Eq. (3) becomes linear in this limit. The steadystate profile for λ ≫ c1/D0 behaves as λ ∼ exp(−x/ξd). In the interval λT < λ < c1/D0 theligand profile is well described by Eq. (6), while for small λ ≪ λT , λ ∼ exp(−x/ξ). As aconsequence of finite D0, the robustness approaches a finite value Rmax = a/ξd for large j0.For ξd < a, this implies R > 1 for large j0 and cells in the target tissue are insensitive tovariations of j0 by a factor of two. The robustness is displayed in Fig. 4 as a function of j0for different values of ξd/a.Finally, we briefly discuss how the transport Eq. (3) is derived from a microscopic model.We denote the number of free ligands in the space between cell n and n + 1 by Ln and thenumbers of receptor bound ligands inside cell n by S(i)n . Similarly, we introduce the numbersS(r)n and S(l)n of receptor-bound ligand on the right and left surface of cell n, respectively,see Fig. 1. The dynamics of these quantities is given by∂tLn = koff(S(r)n + S(l)n+1)− kon(R− S(r)n − S(l)n+1)Ln − edegLn (8)∂tS(r)n = −koffS(r)n + kon(R2− S(r)n)Ln − bintS(r)n +12bextS(i)n (9)∂tS(l)n = −koffS(l)n + kon(R2− S(l)n)Ln−1 − bintS(l)n +12bextS(i)n (10)∂tS(i)n = −bextS(i)n + bint(S(l)n + S(r)n )− bdegS(i)n . (11)On large scales, we describe the ligand profiles by the densities l(x) = Ln/a, s(x) = S(i)n /aand s±(x) = [S(l)n ±S(r)n ]/a, where x = na. Because transport on large scales L is governed bytime scales τL ≃ L2/D which are long compared to the relaxation time on the cellular scaleτa ≃ a2/D, three of the dynamic equations can be adiabatically eliminated. The effectiveEq. (3) for the total ligand density λ = l+s+s+ follows by taking a continuum limit [18]. Acomparison of the full solutions of Eqs. (8)-(11) to Eq. (3) is shown in Fig. 3, demonstratingthat the effective transport Eq. (3) captures the behavior of the microscopic model.In conclusion, we have developed a general theoretical framework for ligand transportand the generation of characteristic ligand profiles via transcytosis. Cell differentiation istriggered at threshold levels of morphogen concentration. The robustness of patterning hasbeen demonstrated experimentally under conditions where Dpp was overexpressed [11, 16,19]. Reliable patterning can be achieved by robustly generating morphogen profiles. Wehave defined the robustness of steady state ligand profiles with respect to the rate of ligandsecretion j0. We find that transcytosis naturally leads to a large robustness of the profile.6Free extracellular diffusion reduces robustness significantly if the diffusion length ξd exceedsseveral cell sizes. The origin of robustness in transcytosis are nonlinearities in the ligandcurrent. A similar mechanism based on nonlinear degradation alone permits the robustformation of gradients in a free diffusion model [6]. Robustness can also be achieved byother nonlinear effects [7].In the general case where both ligand and receptor densities are taken into account, newphenomena appear [18]. In particular ligand currents can be driven by receptor gradients.Furthermore, ligand gradients are accompanied by a gradient in the receptor density as isindeed observed experimentally [11, 20]. In addition, if cells posess a polarity and define adirection in the tissue, transcytosis can lead to directed transport. Finally, it is straightfor-ward to generalize the concepts presented here to higher dimensions.The effects discussed here become important if the diffusion length ξd is smaller than therange ξ ≃ 30a of the gradient. From experiments where endocytosis has been blocked inthe target tissue, we estimate ξd ≤ 2a [11]. The rates of transcytosis have to be sufficientlylarge to guarantee gradient formation within the experimentally observed times of aboutk(0)−1 ≃8 hours. Assuming that the rates konR, koff , bint, and bext are of the order of 1/min,we estimate D ∼ a2/min. This diffusion coefficient is sufficient to generate a gradient overa length ξ ≃ 30a during 8 hours. Numerical solutions to the transcytosis model (see Fig. 3)confirm this estimate. A rate of one per minute is larger than values measured in differentbut related systems [21] but is plausible for intracellular trafficking.The values of the rate constants are difficult to measure while the effective diffusion coef-ficient and the effective degradation rates are more easily accessible in fluorescence recoveryexperiments. Our work suggests that if transcytosis dominates transport, morphogen gra-dients are robust with respect to morphogen overexpression. This can be directly tested infuture experiments.[1] A. M. Turing, Philosophical Transactions of the Royal Society of London Series B 237, 37(1952).[2] M. C. Cross and P. C. Hohenberg, Reviews of Modern Physics 65, 851 (1993), part 2.[3] A. J. Koch and H. Meinhardt, Reviews of Modern Physics 66, 1481 (1994).7[4] A. Gierer and H. Meinhard, Kybernetik 12, 30 (1972).[5] L. Wolpert, Journal of Theoretical Biology 25, 1 (1969).[6] A. Eldar, D. Rosin, B. Z. Shilo, and N. Barkai, Developmental Cell 5, 635 (2003).[7] A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B. Z. Shilo, and N. Barkai, Nature 419, 304 (2002).[8] B. Houchmandzadeh, E. Wieschaus, and S. Leibler, Nature 415, 798 (2002).[9] F. Crick, Nature 225, 420 (1970).[10] N. McDowell, A. M. Zorn, D. J. Crease, and J. B. Gurdon, Current Biology 7, 671 (1997).[11] E. V. Entchev, A. Schwabedissen, and M. González-Gaitán, Cell 103, 981 (2000).[12] T. Y. Belenkaya et al., Cell 119, 231 (2004).[13] M. Kerszberg and L. Wolpert, Journal of Theoretical Biology 191, 103 (1998).[14] S. Pfeiffer, C. Alexandre, M. Calleja, and J. P. Vincent, Current Biology 10, 321 (2000).[15] A. D. Lander, Q. Nie, and F. Y. M. Wan, Developmental Cell 2, 785 (2002).[16] K. Kruse, P. Pantazis, T. Bollenbach, F. Jülicher, and M. González-Gaitán, Development 131,4843 (2004).[17] E. V. Entchev and M. A. González-Gaitán, Traffic 3, 98 (2002).[18] T. Bollenbach et al., to be published.[19] S. Morimura, L. Maves, Y. Chen, and F. M. Hoffmann, Developmental Biology 177, 136(1996).[20] A. A. Teleman and S. M. Cohen, Cell 103, 971 (2000).[21] D. A. Lauffenburger and J. J. Lindermann, Receptors: models for binding, trafficking, andsignalling (Oxford University Press, 1993).[22] We find k(λ) = C+(λ) [bdegbint/[2bext (bext + bint)] + edegkoff/C−(λ)] /κλ. For D0 = 0, wefurthermore have D(λ) = [a2bextbintkoffκrC−(λ)]/[4A(λ)[2κrkoff (bext + bint) + bintC−(λ)]].In these expressions κ = akon, A(λ) =[−4bext(bext + bint)κ2rλ+ (bintκr + bextB+(λ))2]1/2,B±(λ) = koff + κ(λ± r), C±(λ) = bintκr ∓A(λ)± bextB±.8FIG. 1: Schematic representation of transport by transcytosis in a chain of cells of diameter aindexed by n. The rates of ligand-receptor binding and unbinding, internalization and external-ization of ligand-receptor pairs are denoted kon, koff , bint and bext. Degradation of ligand occursinside the cells with rate bdeg and in the extracellular space with rate edeg.FIG. 2: Effective diffusion coefficient D(λ) as a function of ligand concentration λ for D0/a2bdeg =10 (solid line) and D0 = 0 (dashed line). Inset: effective degradation rate k(λ) as a function of λ foredeg = 0 (dashed line) and edeg/bdeg = 5 (solid line). Parameters are: bint/bdeg = bext/bdeg = 3×103,konR/bdeg = 1.1 × 104, koff/bdeg = 7× 102.9FIG. 3: Ligand densities λ(x) in the presence of a source at x = 0 at different times bdegt =0.18, 0.54, 0.9, 1.26 during gradient formation and in the steady state. Lines indicate solutions toEq. (3), while symbols indicate solutions to Eqns. (8)-(11) for comparison. The robustness of thesteady state profile is R ≈ 470. Parameters as in Fig. 2 with D0 = 0, j0/bdegR = 70 and j = 0 atx/a = 50. Inset: steady state ligand profiles for same parameters but j0/bdegR = 7 where R ≈ 0.1.The profile (solid line) is strongly affected by halving (dotted line) or doubling (dashed line) theligand current of the reference state.FIG. 4: Robustness R of steady state ligand profiles as a function of the ligand current j0 fromthe source for different values of the ratio of the diffusion length ξd and the cell size a. Parametersas in Fig. 2.10

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Robust formation of morphogen gradients

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