# Mathematical analysis of a model of Morphogenesis: steady states. J. Ignacio Tello

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3 On the mathematical analysis of a model of morphogenesis by LR the complex ligand-receptor and their respective concentrations by [L], [R] and [LR]. The processes, when the ligand L binds with a receptor R to form the complex LR and the reversal are expressed in the following formula L + R k on k off LR, where k on and k off are the binding and dissociation rate constants. Then, the reaction occurs at rates k on [L][R], and k off [LR]. We assume [R] = R tot [LR], (1) where R tot is the total receptor concentration per unit of extracellular space. Assumption [1] is biologically meaningful only for small intervals of time and reduces the number of equations simplifying the problem. For large intervals of time, degradation of [R] and [LR] has to be considered (see Lander, Nie, Vargas, Wan [7] and J.I. Tello [16]). As we explain in the previous section we consider linear diffusion of [L] with diffusion constant d. Then, [L] satisfies the following equation: where 2 [L] d t x 2 [L] = k onr tot [L] + k on [L][LR] + k off [LR], x >, t >, (2) d 1 11 M 2 s 1 ; k on R tot 1 2 s 1 ; k on 1 6 M 1 s 1 ; k off 1 6 s 1 (see Lander, Nie and Wang [6] and references there). The equation governing the bound receptor dynamics does not include diffusion but the degradation of the complex is introduced in the model. Let k deg be the degradation rate constant, then [LR] satisfies the equation: where t [LR] = k onr tot [L] k on [L][LR] k off [LR] k deg [LR], t >, (3) k deg s 1. We consider that the morphogen is synthesized at x = with rate k syn and that the concentration of [L] goes to as x goes to infinity. Then the boundary conditions for [L] are the following t [L] = k syn k on R tot [L] + k on [L][LR] + k off [LR], x =, t > ; lím x t > ; (4) where k syn

4 J. Ignacio Tello (see Lander, Nie, Wang [5]). The system (2)-(4) is completed with the initial data We introduce the dimensionless variables: [L] = [LR] =, x >, t =. (5) ( ) kon R 1/2 tot t := k on R tot t; x := x; u := k on [L]; d k off v := [LR] R tot and the parameters µ := k deg k off R tot ; λ := k off k on R tot ; ν := k syn k off R tot. Dropping the tildes and replacing the new variables in equations [2]-[5] we get the dimensionless version of the model: u t 2 u = u(1 v) + v, x >, t >, (6) x2 v = λ [u(1 v) v] µv, t x >, t >, (7) with boundary conditions and initial data: u t = ν u(1 v) + v, at x =, t >, (8) lím u(x, t) =, t >, (9) x u(x, ) = v(x, ) =, x. (1) Through the paper, we assume for technical reasons in concordance to experimental data that µ > ν >. (11) λ 3. Steady states We consider the solutions to satisfying the boundary conditions: = 2 φ φ(1 ξ) + φ; x >, (12) x2 = λ [φ(1 ξ) ξ] µξ, x >, (13) = ν φ(1 ξ) + ξ, at x =, lím φ(x) =. (14) x Lemma 1 For every µ, λ, ν satisfying (11) there exists a unique solution (φ, ξ) to (12)- (14). Moreover, φ and ξ are monotone decreasing functions. 4

5 On the mathematical analysis of a model of morphogenesis Proof: Combining (13) with (14) we get ξ() = νλ µ := β < 1, which, when combined with (14), yields the boundary condition φ() = ν + β 1 β = ν(µ + λ) µ νλ := α >. (15) Because of (13), ξ is defined as follows Hence, (12) becomes ξ = φ φ µ, for µ := µ λ. = 2 x 2 φ µ φ ; x >. (16) φ µ Multiply equation (16) by ( φ) + (where ( ) + is the positive part function) and integrate by parts over (, ) to obtain, by (15), that ( φ) + =, i.e φ. We introduce the system of ODE s satisfying the initial datum φ() = α and φ = ζ; (17) ζ = µ φ φ+1+ µ, lím φ(x) = lím ζ(x) =. (18) x x We examine the phase portrait of (17) in the half plain φ. The unique equilibrium is (, ) and has eigenvalues ± µ, so that (, ) is a saddle point. Notice that: φ >, ζ > for φ >, ζ > ; φ <, ζ > for φ >, ζ < ; φ <, ζ < for φ <, ζ < ; φ >, ζ < for φ <, ζ > ; φ >, ζ = for φ =, ζ > ; φ =, ζ > for φ >, ζ = ; φ <, ζ = for φ =, ζ < ; φ =, ζ < for φ <, ζ =, 5

6 J. Ignacio Tello and so there exists a unique orbit which may provide a solution to (17) satisfying lím φ(x) =. x We denote by γ = (γ 1, γ 2 ) this orbit. To conclude the proof we have to prove that γ intersects with φ = α. Multiply (17) by (φ, ζ) to obtain φφ = ζφ, which implies and so i.e. 1 2 (ζ2 ) ζφ = µ φ µ, 1 2 (ζ2 ) φφ = µ φ µ, ( ) d 1 dx 2 ζ2 µφ µ ln(φ µ) =, 1 2 ζ2 (x) µφ(x) µ ln(φ(x) µ) = const. (19) Since γ belongs to the region φ >, ζ < ; there are no periodic orbits and the unique equilibrium is (, ) we have that (19) and (2) imply that lím γ =. (2) x lím γ 1 =, x lím γ 2 =, x then, for every α >, γ intersect the line φ = α and so there exists a unique solution to (12), (13). Lemma 2 There exists a unique solution φ to (16) satisfying the boundary condition φ() = a, for a (, α]. Moreover, for every a (, α], φ a is a monotone decreasing function of x. The proof is similar to that of Lemma 1, therefore we omit the details. We denote by φ a the solution to (16) satisfying φ() = a >, then Lemma 3 φ a C 2,α (Ω) H 1 (Ω) L p (Ω) for 1 p and lím a α φ φ a dx =. 6

7 On the mathematical analysis of a model of morphogenesis Proof: By (19) we deduce that φ a() 2 = 2( µ(a + ln(a µ))) + const <, which combined with the fact φ > (see Lemma 1) we have φ a const. Multiply (16) by φ p a and integrate over (, ) to obtain φ a L p (Ω) H 1 (Ω), for 1 p <. φ a L (Ω) is a consequence of the monotonicity of Φ a, then < φ a a. Since φ a W 1, (Ω) H 1 (Ω) we have that φ a C,δ (Ω) and then d2 dx φ a C,α (Ω) which implies φ a C 2,α (Ω). Let x a be defined as the unique point in (, ) such that φ(x a ) = a (i.e. x a := φ 1 (a)). Notice that lím x a =, (21) a α and (by uniqueness of solutions) we have that Then, φ φ a dx = φ a (x) = φ(x + x a ). φ dx xa φ a dx = φ dx x a α. φ dx φ dx = x a Taking limits when a α in the above equation, and by using (21) we obtain which ends the proof. lím a α φ φ a dx =, References [1] R.A. Adams, Sobolev Spaces. Academic Press, Orlando [2] E.V. Entchev, A. Schwabedissen, M. González-Gaitán, Gradient formation of the TGFß homolog Dpp. Cell, 13 (2) [3] E.V. Entchev, M. González-Gaitán, Morphogen gradient formation and vesicular trafficking. Traffic, 3 (22) [4] M. Kerszberg, L. Worpert, Mechanism for positional signalling by morphogen transport: A theoretical study. J. Theor. Biol. 191 (1998) [5] A.D. Lander, Q.Nie, F.Y.M. Wan, Do morphogen gradients arise by diffusion?. Dev. Cell, 2 (22) [6] A.D. Lander, Q.Nie, F.Y.M. Wan, Internalization and end flux in morphogen gradient degradation. J. Comput. Appl. Math. 19, No 1-2 (26) [7] A.D. Lander, Q.Nie, B. Vargas, F.Y.M. Wan, Agregation of a distributed source morphogen gradient degradation. Studies in Appl. Math. 114 (25)

8 J. Ignacio Tello [8] G.M. Lieberman, Second Order Parabolic Differential Equations. World Scientific, Singapure [9] J.L. Lions, Quelques méthodes de résolution del problemes aux limites non linéaires. Dunod-Gauthier Villars, Paris [1] Y. Lou, Q.Nie, F.Y.M. Wan, Effects of sog DPP-Receptor binding. Siam J.Appl. Math. 65, No 5 (25) [11] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, NJ, [12] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order. Springer, Berlin- New York, [13] J.H. Merking, D.J. Needham, B.D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity. Nonlinearity, 18 (25) [14] J.H. Merking, B.D. Sleeman, On the spread of morphogens. J. Math. Biol. 51 (25) [15] A. Teleman, S. Cohen, Dpp Gradient Formation in the Drosophila Wing Imaginal Disc. Cell, 13, Issue 6 (2) [16] J.I. Tello, Mathematical analysis of a model of Morphogenesis. Submitted. [17] L. Wolpert, Positional information and the spatial pattern of cellular differentiation. J. Theore. Biol. 25 (1969)

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