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# Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies

• Robust morphogen gradient formation is important for embryo development. Patterns of developmental tissue are encoded by the morphogen gradient that drives the process of cell differentiation in response to different morphogen levels. Experiments have shown that tissue patterning is robust with respect to morphogen overexpression. However, the mechanisms for this robust patterning remain unclear. The expansion-repression mechanism, which was proposed for achieving scaling of patterning with organ size, is a type of self-enhanced clearance through a non-local feedback regulation and may contribute to the robustness with respect to morphogen overexpression. In this paper, we study the role of the expansion-repression mechanism in morphogen gradient robustness through a two-equation model with general forms of feedback functions. We prove the existence of steady-state solutions, and, through model reduction and simplification, show that the expansion-repression mechanism is able to improve the robustness against changes in the morphogen production rate. However, this improvement is restricted by the biological requirement of multi-fate long-range morphogen gradient.
Mathematics Subject Classification: Primary: 34B15, 92C15; Secondary: 34B60, 35K57.

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