Robustness of Tissue Patterns*

Transcription

1 MCBU Project II Robustness of Tissue Patterns* June, 2014 Frederic Y.M. Wan Mathematics University of California, Irvine Supported by: NIH Grants R01-GM67247 P50-GM66051

2 Biological Patterning

3 The architecture of the wing margin adult wing anterior posterior wing disc diagram of cells in the wing disc

4 Receptor-Mediated (or the French Flag) Morphogen Gradients

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7 Generic Signaling Pathway Ligand (Dpp) Receptor (Tkv) Output Outside the Inside the cell Signaling Degradation Ligand = Morphogen

8 Diffusion, Reversible Binding & Degradation with Rewable Receptor Extracellular Formulation Finite Dpp Production Region X = -X min Dpp Tkv X = -X min v > 0 L R ω n > 0 D X = 0 k off X = 0 L + R [LR] v = 0 k on ω p > 0 d D k deg k deg X max Sink Degradation X max

9 Diffusion, Reversible Binding and Degradation (with a Finite Region of Morphogen Synthesis) Extracellular Formulation 2 L L = D 2 konlr + koff [ LR] + VL, T X [ LR] = konlr ( koff + kdeg)[ LR], T R = V R k on LR + k off [ LR ] k deg R, T L X = X min : = 0, X = X max : L = 0 T > 0 X ( ) ( ) T = 0 : L = [ LR] = 0, R = R ( X ) X X X s min max

10 Well-Posed of Steady State Problem Theorem (1): There exists a unique set of nonnegative, steady state concentration gradients {L(X ), [LR(X )], R(X )} characterized by a nonlinear two-point BVP for L(X ). Theorem (2): The unique steady state solution is linearly stable. (via a nonlinear eigenvalue problem). A.D. Lander, Q. Nie & F.Y.M. Wan, Developmental Cell Vol. 2, 2002, Math BioSci & Engrg Vol. 2, 2005,

11 Why do we have or need Glassbottom boat (Gbb) Wingless (Wg) Hedgehog (Hh) Syndecan (Sdn) Dally and Dally like (Dlp) Notum Dfrizzle (Dfz) Thickvein (Tkv) Glypicans Heparan Sulfate Proteoglycans (HSPG) Pentagone (Pent) (and many more)

12 Processes Affecting Ligand Transport and Ligand-Receptor Interactions (Ligand = Morphogen) membrane bound inhibitors down- regulation desensi- tization co-receptors receptor synthesis ligand depletion soluble inhibitors/ activators transcellular transport diffusion sequestration processing secretion ligand synthesis

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14 Dpp, a morphogen of the TGFβ superfamily, is closely related to BMPs 2 & 4

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16 Steady-state gradient shape is determined by two parameters ! too steep too shallow 0.6 Fractional Receptor Occupancy ([LR]/R tot ) 0.2!=66.7!=25!= " distance (µm) ψ = x 2 max k deg k on R tot D (k off + k deg ) β = v R tot k deg

17 Dpp Synthesis Rate as a Function of Temperature Shaohua Zhou (Lander s Lab)

18 b( x) = [ LR] / R in 0 Robustness v 2v Δx 0 d Δx 1 x

19 Steady State Gradient Shape in Parameter Space Want to achieve More useful gradients in a broader parameter region Higher level of gradient robustness with respect to environmental/systems pertubations Need to consider additional biological activities: Feedback control/regulation? (neither necessary nor sufficent) Non-receptor/inhibitor? Co-receptor? Others?

20 In principle, we may choose Robustness [ Δx] 1 1 rms 2 Rv = = [ Δx] db 1 d 1 d b 0 0 d b d as a quantitative measure of robustness. But we will choose to define instead [ Δx] 1 1 x( b ) d x( b ) d b b 4 / 5 rms 2 Rv = [ Δx] db b1/ 5 1/5 1/5 4/5 1/5 b

21 Generic Signaling Pathway Ligand (Dpp) Nonreceptor (Dlp) Receptor other proteins Degradation Outside the Inside the cell Signaling Degradation Output

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23 Non-receptors and Co-receptors Non-receptors bind with morphogens but do not signal Syndecan (Sdn) Dally Dally-like (Dlp) Sog (Sog) Glypicans (such as Dally and Dlp) Heparan Sulfate Proteoglycans MMT-7 (Mus musculus hypothetical protein?) etc. Co-receptors bind with the Dpp and signal

24 Glass bottom boat (Gbb) Wingless (Wg) Hedgehog (Hh) Syndecan (Sdn) Dally and Dally-like (Dlp) Chordin Arrow (Arr) Dfrizzle (Dfz) Thickvein (Tkv) Pentagone (Pent) May be there is a need for Short gastrulation gene (Sog) Glypicans Heparan Sulfate Proteoglycans (HSPG) angiogenesis, blood c., tumor m. (and many more)

25 Bridging Inhibition and Ectopic Signaling Need feedback to trigger inhibiting mechanism for signal reduction Investigated a simple model for a proof of concept study A spatially uniform negative feedback on ligand synthesis rate v L Many other more realistic mechanisms are being investigated

26 Other Types of Feedback Mechanisms Ligand degradation rate constant ( + ) Ligand binding rate constant ( ) Receptor mediated ligand degradation rate constant ( + ) Nonreceptor synthesis rate ( + ) Nonreceptor degradation rate constant ( ) Ligand mediated nonreceptor degradation rate constant ( + )

27 Spatially Nonuniform Feedback Mechanisms Ligand binding rate constant ( ) Receptor mediated ligand degradation rate constant ( + ) Nonreceptor synthesis rate ( + ) Nonreceptor degradation rate constant ( ) Ligand mediated nonreceptor degradation rate constant ( + )

28 Other Factors to Improve Robustness? Multiple non-diffusing Inhibitors Diffusing non-receptors (such as Sog) Co-Receptors Mutation Endocytically Impaired Clones

29 Inhibitors of DPP noggin (NOG) leads to the blocking of endogenous BMP signaling chordin (Chd) antagonizes BMP signaling by blocking binding to their. follistatin (FST, which binds BMP-7 and BMP-2 reversibly short gastrulation (Sog) binds directly with Dpp.

30 Other Inhibiting Mechanisms However, there are also inhibitors of morphogen signaling that achieve the same outcome by different biological processes from those of non-receptors. Among these are wingless (Wg), which down-regulates Dpp receptor frizzle 2 (Fz2) expression, Daughter against Dpp (Dad), which down-regulates the Dpp target gene optomotor-blind (Omb), and PAI-1 which induces receptor-mediated internalization and degradation of urokinase).

31 Proposition: Negative feedback on receptor synthesis rate is neither sufficient nor necessary for a robust gradient M. Khong & F.Y.M. Wan: Negative feedback in morphogen gradients, Frontiers in Applied Mathematics, (Proc. 2 nd International Symposium, June, 2006, Ed. D.Y. Hsieh, et al.) World Scientific, NJ, 2007, pp

32 Positive Results Robustness of Signaling Gradient in Drosophila Wing Imaginal Disc J. Discrete & Cont. Dyn. Syst., Series B 15 (3), 2011, (with J.-Z. Lei, A.D. Lander and Q. Nie) A New Approach to Feedback for Robust Signaling Gradients, Studies in Appl. Math, 2014, to appear (with T. Kushner and A. Simonyan) *Supported by NIH R01-GM67247 NIH P50-GM76516 NSF DMS

33 A Great Deal of Research to be Done!

34 Diffusion, Reversible Binding and Degradation with Receptor Internalization and Renewal!L!T = D!2 L!X! k LR + k [LR] +V (X,T ), 2 on out off out L![LR] out!t![lr] in!t = k on LR out! (k off + k in )[LR] out + k out [LR] in, = k in [LR] out! (k out + k deg )[LR] in,!r out!t = k off [LR] out! (k p + k on L)R out + k q R in,!r in!t =V R (X,T )! (k q + k g )R in + k p R out, for 0 < X < X max and T > 0

35 Negative Feedback on Receptor Synthesis Rate

36 M. Khong and F.Y.M. Wan Negative feedback in morphogen gradients, Frontiers of Applied Mathematics, Ed. D.-Y. Hsieh, et al., World Scientific, 2007,

37 Robustness Simulation

38 Additional Biological Activities Non-receptor/inhibitor sufficient (from simulations) A.D. Lander, Q. Nie & F.Y.M. Wan Co-receptor? Others?

39 Additional Biological Activities Feedback control/regulation on receptor synthesis rate Numerical Simulations not sufficient For robustness (nor necessary) R v L 0 /(K m η R )

40 R v Effect of Non-receptor L 0 /(K m η R ) [LR] in R v L 0 /(K m η R ) x