Introduction

• This paper introduces a new model for regenerating complex biological shapes using the concept of cell memory.
• Many organisms (like planaria and axolotls) can regrow lost parts, which inspires this research.
• The key question is whether regeneration uses only the current signals in cells or also the memory of the original structure.

Model of Regeneration Based on Cell Memory

• Cells are treated as points on a plane that both send and receive signals.
• Each cell produces a signal (u) that spreads out and weakens with distance.
• Before any damage, each cell has an “ideal” or “old” signal value (u*) that represents the correct, original state.
• After damage (amputation), cells start receiving a new signal; the difference (u* − u) triggers regeneration.
• Analogy: Imagine a cake missing a layer. The baker uses the original recipe (memory) to add the correct layer until the cake is complete.

Signal Distribution and Geometry

• Cells are arranged in a two-dimensional grid, forming a shape with a specific signal distribution.
• Cells at the boundary receive less signal because they have fewer neighboring cells.
• When part of the structure is removed, the remaining (control) cells have two signals:
  • The old signal from before the removal.
  • The new signal produced after the damage.
• The difference between these signals near the cut acts as a cue to start cell division and growth.

Algorithm of Regeneration

• New cells are added one by one in discrete time steps.
• Placement rules for new cells include:
  • Cells must be placed on grid nodes adjacent to cells at the damaged edge (ensuring continuous growth).
  • After adding a cell, the new signal in the control cells must not exceed the old signal.
  • The position chosen is the one that minimizes the difference between the old and new signals.
• Metaphor: It is like adding a puzzle piece—each new piece is carefully placed so that the overall picture matches the original.

Nonlinear Diffusion and Parameter Effects

• The spread of the signal can be described by diffusion equations, which may be linear or nonlinear.
• A key parameter is the decay rate (n) in the function f(d) = 1/dⁿ:
  • If n is too high or too low, the regeneration process may be inefficient or incorrect.
• A small threshold (epsilon) is used to ensure that the new signal closely approximates the old signal before growth stops.

Regeneration in Different Shapes and Nonconvex Domains

• The model works best for convex domains, where all points on the boundary are directly connected.
• For nonconvex domains, measuring distances in a straight line may not accurately represent the actual tissue path, making regeneration more challenging.
• Examples in the paper show regeneration in rectangular, elliptical, and even letter-shaped domains.
• Limitation: If the remaining domain is too large or irregular, control cells may not correctly interpret the signals, leading to abnormal regeneration.

Discussion and Implications

• The model is based on a cell memory mechanism where cells compare the stored (old) signal with the current (new) signal and then produce a corrective signal to stimulate growth.
• Regeneration stops when the new signal matches the old signal—this is the target morphology.
• This quantitative model helps explain how organisms can precisely rebuild lost structures.
• Compared to reaction-diffusion (Turing) models that require the interaction of multiple chemicals, this approach uses a single signal with memory.
• Potential applications include understanding wound healing, embryonic development, and unusual cases such as two-headed regeneration in planaria.
• Limitations include sensitivity to small changes and parameters that may differ from actual biological values.
• Future work may involve incorporating different cell types and long-range signals to more accurately mimic complex regeneration.

引言

• 本文提出了一种基于细胞记忆的新模型，用于再生复杂的生物形态。
• 许多生物（如涡虫和蝾螈）能够再生失去的部位，这为该研究提供了灵感。
• 关键问题在于：再生是否仅依赖于细胞当前接收的信号，还是也利用了原始结构的记忆？

基于细胞记忆的再生模型

• 将细胞视为平面上的点，每个细胞既发送也接收信号。
• 每个细胞产生的信号 (u) 向外扩散，并随距离衰减。
• 在损伤前，每个细胞都有一个“旧”信号值 (u*)，代表正确或理想的状态。
• 当部分结构被切除后，细胞接收到新的信号；旧信号与新信号之间的差异 (u* − u) 刺激了再生过程。
• 比喻：就像蛋糕缺了一层，烘焙师依靠原始食谱（记忆）逐步补上缺失的部分，直到蛋糕恢复原状。

信号分布与几何形状

• 细胞排列在二维网格中，形成具有特定信号分布的形状。
• 边缘细胞由于邻居较少，接收到的信号比内部细胞要弱。
• 当部分细胞被移除后，剩余（控制）细胞拥有两种信号：
  • 旧信号：来自原始状态。
  • 新信号：受损后产生的新信号。
• 切口处信号的差异促使细胞分裂和新细胞的生长。

再生算法

• 新细胞按照离散的时间步骤逐个添加。
• 新细胞的放置规则包括：
  • 新细胞必须放置在网格节点上，并且与切除区域的细胞相邻，以确保生长的连续性。
  • 每次添加后，控制细胞的新信号不能超过旧信号。
  • 选择能够最小化旧信号与新信号差异的位置。
• 比喻：就像拼图，每添加一块拼图，都需要确保它能完美地契合原有的拼图，使整体图案恢复原貌。

非线性扩散与参数影响

• 信号扩散可以用扩散方程描述，可能是线性也可能是非线性的。
• 在衰减函数 f(d) = 1/dⁿ 中，参数 n 非常关键：
  • 若 n 值过高或过低，再生过程可能会失效或不完整。
• 使用一个小的阈值 (epsilon) 来保证新信号与旧信号足够接近，这起到了判断是否停止生长的作用。

不同形状及非凸区域的再生

• 该模型最适用于凸形状，即所有边界点都能直接连接的区域。
• 对于非凸形状，由于直线距离不能准确反映组织内部的实际路径，再生过程会更加困难。
• 论文展示了矩形、椭圆甚至字母形状的再生实例。
• 限制：如果剩余区域过大或形状太不规则，控制细胞可能无法正确解读信号，导致再生异常。

讨论与启示

• 细胞记忆机制：细胞会将存储的旧信号与当前的新信号进行比较，并据此产生促进生长的信号。
• 目标形态：当新信号与旧信号一致时，再生停止，表明原始形态已被恢复。
• 该模型提供了一种量化的方法来解释生物体如何精确地重建丢失的结构。
• 与图灵模式相比：反应扩散模型需要多种化学物质的相互作用，而此模型仅依赖单一信号及其记忆。
• 潜在应用包括理解伤口愈合、胚胎发育以及如双头再生等异常现象。
• 局限性在于模型对微小变化非常敏感，且参数可能与真实生物系统存在差异。
• 未来的研究方向可能包括引入不同的细胞类型和长距离信号，以更精确地模拟复杂的生物再生过程。