Introduction: Bioelectric Networks in Regeneration

• Cells use electrical signals—voltage differences across their membranes—to communicate during development and regeneration.
• These bioelectric signals help coordinate how cells form complex tissues and organs, even after injuries.
• Planarian flatworms are an ideal model because they can regrow an entire body from a small fragment.

Electrodiffusion Hypothesis and Model Overview

• The model centers on a charged molecule called a morphogen (assumed to be negative) that establishes an electrical gradient along the worm.
• Two processes set up this gradient: electrical drift (movement under the influence of an electric field) and diffusion (spreading from high to low concentration).
• Cells are connected by gap junctions (GJs), which act like tiny wires that let electrical signals pass between them.

Key Parameters and Their Roles

• num_cells: Total number of cells from the head to the tail of the worm.
• kM and N: Parameters in the Hill model that determine how the morphogen’s concentration affects the opening of ion channels. (The Hill model describes how a change in concentration leads to a switch-like response; think of it as a dimmer switch for cell signals.)
• GJscale: A scale factor for gap junction conductivity. Too high and the system “short-circuits” (smears out voltage differences); too low and cells act independently, forming local islands of voltage instead of a continuous gradient.
• ZM: The valence (charge) of the morphogen. A higher ZM increases the electrical force on the molecule, similar to how a stronger magnet pulls metal objects more forcefully.
• sgd: The time constant for the generation and decay of the morphogen. It tells how quickly cells reach a steady concentration.
• sspread: The time constant for electrodiffusion; it indicates how fast the morphogen spreads along the worm.

Loop Gain and Gradient Formation

• Loop gain is the amplification factor that describes how a small initial difference in morphogen concentration can be magnified into a full-blown gradient.
• If loop gain is greater than 1, a tiny difference grows into a strong gradient essential for proper regeneration.
• If it is less than 1, the small differences fade away and the gradient collapses.
• Loop gain depends mainly on the Hill model cooperativity (N) and the morphogen’s charge (ZM).

Effects of Gap Junction Conductivity (GJscale)

• If gap junctions are too conductive (high GJscale), voltage differences (DVmem) across the worm are reduced, leading to a “short-circuit” effect where no gradient forms.
• If gap junctions are not conductive enough (low GJscale), cells operate too independently, resulting in multiple local voltage islands instead of a single head-to-tail gradient.
• An optimal range of GJscale exists that supports proper global communication; this range must adjust (allometric scaling) as the worm grows.

Time Constants: Balancing sgd and sspread

• For a robust gradient, the time for morphogen generation/decay (sgd) and the time for its spread via electrodiffusion (sspread) must be balanced.
• If generation and decay happen too quickly compared to diffusion, the system resets before a gradient can be established.
• If they are too slow, the gradient may not be reinforced in time.
• The ideal “Goldilocks” scenario is when sgd and sspread are approximately equal, allowing the gradient to form and stabilize.

Simulation Results and Key Findings

• Simulations across thousands of parameter sets reveal that successful regeneration depends on a careful balance of all factors.
• High Hill-model cooperativity (high N) can maintain a gradient in a full worm but fails in small fragments because its response is too steep.
• Lower cooperativity paired with a higher morphogen valence (ZM) produces robust regeneration across various fragment sizes.
• Allometric scaling is necessary: as the worm increases in length, gap junction properties (GJscale) must be adjusted to maintain effective communication.
• Only specific parameter ranges allow the system to reliably form and maintain a gradient needed for regeneration.

Discussion and Conclusions

• The study demonstrates that electrodiffusion can create stable and robust bioelectric gradients under the right conditions.
• This mechanism is crucial for regeneration and offers design principles that might be applied to synthetic bioengineering projects aimed at self-patterning tissues.
• Key predictions include: the morphogen should have a valence greater than 1; gap junction density must scale with organism size; and ligand-controlled ion channels should have low cooperativity to support robust regeneration.
• The model is computational, so further experimental work is needed to confirm that these mechanisms operate in living organisms.
• The concepts may also extend to other systems (for example, the coordinated heartbeat in the human heart requires similar allometric scaling of gap junctions).

Future Work

• Further experimental verification is required to determine if electrodiffusion is the primary mechanism in planarian regeneration.
• Other models—such as reaction-diffusion and axonal transport—and their combination with electrodiffusion should be explored.
• Evolutionary algorithms may be used to design synthetic tissues with robust, self-patterning capabilities based on these principles.
• Investigations in other organisms, including mammals, could reveal whether these design principles extend beyond planaria.

引言：再生过程中的生物电网络

• 细胞通过膜两侧的电压差（电信号）进行通信，这对于胚胎发育和再生非常重要。
• 这些生物电信号帮助协调细胞如何组建复杂的组织和器官，即使在受伤后也能恢复正常结构。
• 水蠕虫（扁虫）是理想的研究模型，因为它们可以从一小块断片中再生出完整的身体。

电扩散假说及模型概述

• 模型聚焦于一种带电分子，称为形态发生素（在此模型中假设为负电荷），它在水蠕虫体内形成电梯度。
• 电梯度的形成依靠两种过程：电漂移（在电场作用下的定向移动）和扩散（由高浓度向低浓度的自然扩散）。
• 细胞通过缝隙连接（GJ）相互联结，这些连接就像微小的电线，将电信号传递给相邻细胞。

关键参数及其作用

• num_cells：表示从头到尾的细胞总数。
• kM 和 N：Hill模型中的参数，决定了形态发生素浓度如何影响离子通道的开启。（Hill模型类似于调光开关，描述浓度变化如何产生开关效应。）
• GJscale：代表缝隙连接的导电强度。过高会导致“短路”（电压差被平滑掉），过低则会使细胞独立运作，形成多个局部电压岛而非连续梯度。
• ZM：形态发生素的价数；较高的ZM会增强电场对该分子的作用，就像更强的磁铁吸引金属一样。
• sgd：形态发生素的生成与衰减时间常数，决定了细胞达到稳态浓度的速度。
• sspread：电扩散的时间常数，表示形态发生素沿水蠕虫传播的速度。

环路增益与梯度形成

• 环路增益是放大因子，描述了初始微小浓度差如何被放大成明显梯度的过程。
• 当环路增益大于1时，微小差异会逐渐放大，形成再生所必需的强梯度；反之则会消失。
• 环路增益主要取决于Hill模型的协同效应（N值）和形态发生素的价数（ZM）。

缝隙连接导电性（GJscale）的影响

• 如果缝隙连接过于导电（GJscale过高），细胞间的电压差（DVmem）会被削弱，导致梯度无法形成，出现“短路”现象。
• 如果导电性太低（GJscale过低），细胞会独立决策，形成多个局部电压岛，而不是形成统一的头尾梯度。
• 存在一个最佳的GJscale范围，既能保证有效的全局通信，也会随着水蠕虫体型的增长而相应调整（全身比例缩放）。

时间常数：平衡sgd与sspread

• 为了形成稳固的梯度，形态发生素生成与衰减的时间（sgd）和电扩散的时间（sspread）必须保持平衡。
• 如果生成和衰减速度过快，相对于扩散来说，系统会在梯度形成之前就被重置；反之，如果过慢，梯度可能无法及时巩固。
• 最佳状态是两者大致相等，这种“刚刚好”的状态有助于梯度的形成与稳定。

模拟结果及主要发现

• 通过大量参数组合的模拟，发现成功再生依赖于各参数间的精妙平衡。
• 高Hill协同效应（高N值）虽然能在完整水蠕虫中维持梯度，但在小断片中因过于敏感而效果不佳。
• 较低的协同效应配合较高的形态发生素价数（ZM）能在不同断片大小中实现稳健的再生。
• 全身比例缩放必不可少：随着水蠕虫体长增加，缝隙连接属性（GJscale）必须调整以维持有效通信。
• 只有在特定参数范围内，系统才能可靠地形成并维持再生所需的梯度。

讨论与结论

• 研究表明，在合适条件下，电扩散机制能够形成稳定且稳健的生物电梯度，这对再生至关重要。
• 这一机制为设计具有自我模式形成能力的人工组织提供了理论基础，可应用于生物工程领域。
• 主要预测包括：形态发生素应具有大于1的价数；缝隙连接密度需随生物体大小变化；以及受配体控制的离子通道应具有低协同效应以实现稳健再生。
• 模型目前基于计算机模拟，实际生物机制仍需通过实验验证。
• 该原理也可能适用于其他系统，如心脏组织中需要类似的缝隙连接比例调整来协调心跳。

未来工作

• 进一步的实验验证将确认电扩散是否为水蠕虫再生的主要机制。
• 探索其他模型（如反应扩散、轴突运输）以及它们与电扩散结合在稳健模式形成中的作用。
• 利用进化算法设计出具有理想稳健再生特性的人工生物电组织。
• 研究这些机制在其他生物体（包括哺乳动物）中的适用性，以检验这些设计原理是否普遍存在。