标题： 形态发生素梯度的自相似动力学 显示英文标题

标题： Self-similar dynamics of morphogen gradients

摘要： 我们发现了一类自相似解，在非线性模型中描述形态发生素梯度的形成，这些分子浓度场作为发育组织中细胞分化的空间调节因子。 这些模型考虑了局部产生的化学信号的扩散和自诱导降解。 当生产开始时，系统中的信号浓度等于零。 我们发现，在信号生产强度无限大的极限情况下，该问题的解由稳态浓度分布和扩散相似变量的函数的乘积给出。 我们推导出该函数满足的非线性边界值问题，并使用变分方法证明在自然设定下该问题有一个唯一解。 利用分析建立的解的渐近行为，我们通过射击法数值构造了这些解。 最后，我们证明所得到的解可以很容易地用简单的解析表达式近似，从而提供了对形态发生素梯度形成的重要非线性模型的准确全局表征。 我们的结果展示了分析方法在研究生物物理过程的非线性模型中的强大能力。 显示英文摘要

摘要： We discovered a class of self-similar solutions in nonlinear models describing the formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differention in developing tissues. These models account for diffusion and self-induced degration of locally produced chemical signals. When production starts, the signal concentration is equal to zero throughout the system. We found that in the limit of infinitely large signal production strength the solution of this problem is given by the product of the steady state concentration profile and a function of the diffusion similarity variable. We derived a nonlinear boundary value problem satisfied by this function and used a variational approach to prove that this problem has a unique solution in a natural setting. Using the asymptotic behavior of the solutions established by the analysis, we constructed these solutions numerically by the shooting method. Finally, we demonstrated that the obtained solutions may be easily approximated by simple analytical expressions, thus providing an accurate global characterization of the dynamics in an important class of non-linear models of morphogen gradient formation. Our results illustrate the power of analytical approaches to studying nonlinear models of biophysical processes.