Long Range Communication via Gap Junctions and Stress in Planarian Morphogenesis: A Computational Study

Abstract

Developmental morphogenesis reliably builds species-specific large-scale anatomies. Crucially, regulative development and regeneration after diverse and unpredictable damage illustrate that groups of cells deploy context-sensitive, homeostatic algorithms that can navigate anatomical state space in flexible ways. While the molecular machinery necessary for this process is increasingly becoming characterized, the algorithms that guide growth and form are still poorly understood. To drive progress in regenerative medicine and bioengineering, constructive models need to be formulated and tested that show the kinds of information exchange sufficient for specific morphogenetic competencies. Here, we propose a computational model of planarian regeneration that relies on cell-cell communication via gap junction channels and the use of stress as a driver of homeostatic change. We simulate key experiments in the planarian model system and show that this framework is sufficient to demonstrate the observed regenerative behavior of planaria under bioelectric perturbation. This model suggests testable hypotheses in vivo, and it provides a framework for design strategies for control circuits in synthetic bioengineering.

Introduction

Embryogenesis, regeneration, and remodeling of metamorphosis are remarkable processes in which large numbers of cells create and repair a species-specific target morphology.^1,2 Understanding this process is crucial not only for basic evolutionary developmental biology, but also for the discovery of efficient interventions in regenerative medicine.

The ability to specify what shape a collective of cells will build is at the heart of problems such as birth defects, regeneration from traumatic injury and degenerative disease, cancer, and aging. Thus, it is imperative to gain a quantitative understanding of the dynamics that allow individual cells to cooperate toward a reliable large-scale outcome.

One of the most important and least understood processes is the cessation of growth when the correct pattern is complete. For example, animals such as axolotls can restore perfect copies of amputated limbs, tails, eyes, jaws, and other organs³: rapid growth and remodeling ceases precisely when the correct anatomical structure is completed.

Morphogenesis is not simply a hardwired set of steps but is context-sensitive and able to handle unexpected situations: regardless of where a limb is amputated, salamanders can regenerate exactly what is missing, no more and no less. Not only can this process restore structures from standard configurations (e.g., a normal limb amputated at the elbow), but it can also achieve the desired goal from evolutionarily unexpected starting configurations.

For example, tadpoles with scrambled positions of craniofacial organs still make normal frogs,⁴ because the eyes, nostrils, and mouth move around in a novel path from abnormal initial positions to reach a correct frog anatomy. This capacity has been cast as a kind of anatomical homeostasis, progressively reducing the error between the current state and a specific anatomical setpoint (the target morphology).^1,5

How do cellular collectives navigate morphospace⁶ to reach the correct anatomies despite perturbations? A number of things are known about this process. First, the necessary endogenous communication between cells occurs via chemical, biomechanical, and bioelectrical signals.^7,8 Second, the genome does not provide an actual blueprint of the anatomy, but instead specifies the micro-level hardware (proteins) of each cell; it is the developmental physiology that processes the information required to implement morphogenesis.^9–11

Third, the setpoint toward which cells build is specified, in part, by bioelectrical prepatterns—spatiotemporal distributions of resting potential,¹² established by ion channels and shaped by electrical synapses known as gap junctions. Indeed, specific manipulation of ion channel and pump function has been shown to induce ectopic organs,¹³ duplicate the primary body axis,^14,15 repair defects of the brain,¹⁶ trigger regeneration of appendages under non-regenerative conditions,¹⁷ prevent oncogene-driven tumorigenesis,^18,19 and induce morphologies appropriate to other species without genomic editing or transgenes.²⁰

While the molecular components necessary for development and regeneration are increasingly characterized,^21–25 many deep problems (such as the expected morphology of chimeric organisms) remain,²⁶ because we do not yet understand the algorithms that are sufficient for cells to collaborate toward complex anatomical outcomes. It is necessary to design and test models of minimal circuits that would recapitulate system-level morphogenetic repair processes. Here, we specifically focus on bioelectric components of such circuits, in planarian flatworms—a model system well known for its remarkable morphogenetic control²⁷ and its reliance on biophysical mechanisms.^28–30

In planaria, not only is the number and location of heads specified by a gradient of resting potential,³¹ but also once reset to a two-headed pattern, fragments continue to regenerate as two-headed with no further manipulation, despite the wild-type genome.^32,33 The ability to stably re-specify the encoding of information guiding cellular morphogenesis suggests that powerful interventions may be identified that induce desired repair and remodeling in biomedical contexts by manipulating bioelectric signaling (akin to the neural decoding and optogenetic actuation in neuroscience^34–37).

All of these approaches, as well as the understanding of the evolution and self-repair of morphologies, require a better understanding of emergent minimal bioelectric dynamics sufficient for growth control. Thus, here we computationally model bioelectric communication between cells to uncover dynamics sufficient for the establishment of complex morphologies. Such models have been analyzed earlier, but they did not include detailed spatial analyses of gap junctional paths or a stress (error) parameter.^38–40 In our model, cells use a combination of morphogen signaling and the propagation of stress—an explicit parameter reflecting the distance between the current shape and the encoded target morphology.

We show that this is sufficient to recapitulate two basic results from the classic planarian experimental corpus,⁴¹ including a single cut, and amputation in the presence of a gap junction inhibitor. Our model is a possible springboard for understanding the anatomical homeostasis of cellular collectives and for designing novel bioengineering constructs such as self-repairing living machines.

Models and Methods

We are interested in how long-range communication paths control and guide the collective regeneration processes in (morphological) space and time. It is now known that gap junctions and ion channels establish compartment boundaries for organs,^42–50 and that alterations of bioelectric properties influence the shaping of organs such as the brain, as well as control of the transition from normal tissue to tumorigenesis, at long range.⁵¹ Thus, we built a model to better understand the control of morphogenesis during regeneration by bioelectric dynamics propagating through tissues. An overview of our model approach is depicted in Figure 1.

FIG. 1.

Model overview: (A) Our model hinges on three primary forces that drive the dynamics of morphogenesis: intracellular stress-voltage coupling, intercellular stress signaling, and intercellular membrane voltage propagation through gap junctions. Each cell consistently upholds specific stress $(S)$ and membrane voltage $(V)$ setpoints and maintains morphogen concentration $(M)$ . When external events, such as tissue damage or loss, occur, they are detected at the boundary, which, in turn, triggers a cascade of stress excitations. This means stress is propagated to the neighboring cells. Elevated stress levels within cells cause their membrane potentials to shift toward depolarized states, which, in turn, provide feedback on cells' stress states. It is well understood that these depolarized states can initiate downstream dynamics leading to cell proliferation (e.g., in cancer and embryogenesis). Both heightened stress levels and altered cell membrane potentials are communicated to neighboring cells. The foundation of stress signaling remains to be clarified, but it is currently modeled as a diffusion process between immediate neighbors. Over time, these signaling pathways can form more complex structures, referred to as stress networks, within tissues and organisms. We model intercellular stress propagation as a q-state Potts model that reduces in the limit $q \to 1$ to a (directed) bond percolation process, as illustrated in Figure 2. Excited membrane potentials synchronize between neighboring cells via gap junctions. This dynamic is represented on the cell network where cells serve as nodes and gap junctions as resistive edges. Propagation of the excited membrane potentials relies on the Laplacian propagator, detailed in Potential propagation via Gap Junctions. We hypothesize that high morphogen concentrations $(M)$ decrease membrane potentials, which has an impact during the proliferation phase; see Figure 3. (B) The interplay of intracellular membrane voltage $(V)$ and stress level $(S)$ is depicted as interconnected PI controllers, $Π_{V}$ and $Π_{S}$ respectively. The control signals for stress and voltage, $μ_{s}$ and $μ_{v}$ are linked through the function $f (μ_{s}, μ_{v}, α)$ , where $α$ stands for the coupling constant. In addition, stress level in an individual cell is influenced by the stress levels in its immediate surrounding cells, represented as $Ω$ . Each controller within a cell i maintains a setpoint $Λ_{x}$ with $x \in \{s, v\}$ . The error $U_{x} (t) = Λ_{x} - y_{x} (t)$ , that is, the difference between the setpoint and the read-out signal $y_{x} (t)$ is the function to be minimized in each cell after an external intervention such as tissue loss. We hypothesize that proliferation stops when the stress network relaxes below an overall error, that is, the sum of stress setpoint deviations falls below a threshold. Therefore, stress levels cease to affect membrane potential changes. Details are outlined in coupled stress-voltage controllers. (C) Planarians are modeled as cell networks. We abstract the cell shapes, retaining only the spatial distribution of their centers, and model each cell center as a node, connecting it with an edge to its neighboring cells. We represent the cell network as an adjacency matrix $A_{i j} = 1$ iff cell i is connected to cell j and $A_{i j} = 0$ otherwise. The adjacency matrix A is symmetric by construction. (D) We show membrane voltage and stress profiles across the anterior-posterior axis. We initialize each in silico experiment with a fixed membrane potential and stress profile. We generate the membrane potential profile by fixing high potentials at random anterior nodes and low potentials at posterior nodes. We then apply the procedure described in Potential propagation via Gap Junctions. For the morphogen dispersion along the A-P axis, we fix morphogen concentrations for a fixed set of randomly chosen nodes at the posterior. We then calculate morphogen's concentration at each cell by the power-law decay formula $c = a d^{k},$ where a is the proportionality constant and k is the decay exponent. Suitable values for a and k are $a \approx 3$ and $k \approx - 0.1$ , respectively. The Euclidean distance d from each cell to the source nodes informs the morphogen's concentration at that cell. PI, proportional integral.

We describe the stress network generation in Figure 2 as follows:

FIG. 2.

Stress network generation. (A–D) Different stages of stress propagation starting from a cut in the middle of the planarian. Propagation is modeled as a directed bond percolation process. In a directed bond percolation framework, the nodes of the planarian framework represent cellular entities, whereas edges delineate deterministic communication pathways. The transmission of stress along these conduits is probabilistically governed by parameter p. As in real gap junctions, each edge is rendered “permeable” with a likelihood p and “impermeable” with a converse probability $1 - p .$ Effective intercellular stress propagation evolves via these permeable conduits. As the stress percolates, certain paths may become occluded due to impermeable edges, engendering potential segmentation of cellular clusters. This paradigm elucidates the stochastic mechanics underpinning stress distribution within planarian cellular architectures. (E) Schematic diagram showcasing a percolation process from left to right. Connected red cells have an open communication path exchanging stress-related information. For example, cell number 4 communicates stress to its immediate neighbors, specifically cells 1, 5, and 6. Subsequently, these neighboring cells—1, 5, and 6—respond by providing feedback directly to cell number 4. We denoted this neighborhood in Figure 1 as $Ω .$ Dashed edges indicate forbidden configurations. We relate the probability to an inverse temperature indicating a stress factor; see the subsection Stress Network Based on Potts Model given next. (F) Fraction of nodes involved, as a function of probability.

Coupled stress-voltage controllers

We consider two coupled proportional integral controllers for stress and membrane voltage (Fig. 1B). The control action from one controller influences the other and vice versa through a constant factor, α.

The discrete errors are given by: $\begin{matrix} U_{s} [k] = Λ_{s} [k] - y_{s} [k], \\ U_{v} [k] = Λ_{v} [k] - y_{v} [k] . \end{matrix}$

The discrete control action for the stress controller without coupling is given by: $μ_{s, u n c o u p l e d} [k] = R_{s}^{p} U_{s} [k] + R_{s}^{I} T_{s} \sum_{n = 0}^{k} U_{s} [n]$ and $μ_{v, u n c o u p l e d} [k] = R_{v}^{p} U_{v} [k] + R_{v}^{I} T_{s} \sum_{n = 0}^{k} U_{v} [n]$ for the voltage controller respectively. k represents discrete time steps.

We introduce the coupling between the two as follows. $f (μ_{s}, μ_{v}, α) = α \cdot μ_{x} [k]$ . The stress control signal is enhanced by the stress signals from l environmental cells, and we label the coupling as λ. We denote the total coupling between the two controllers as:

We arrive at first-order difference equations of the following kind:

In this scenario, the two controllers are additively coupled. Other scenarios are possible. However, we show that with this simple coupling function we get a rich set of complex state space trajectories.

In the continuous limit, we can derive a coupled system of integro-differential equations: $\begin{matrix} τ_{s} \frac{d y_{s} (t)}{d t} + y_{s} (t) = R_{s}^{p} U_{s} (t) + R_{s}^{I} \int_{0}^{t} U_{s} (τ) d τ + α \cdot μ_{v} (t) + λ \int_{Ω} s d (s) \\ τ_{v} \frac{d y_{v} (t)}{d t} + y_{v} (t) = R_{v}^{p} U_{v} (t) + R_{v}^{I} \int_{0}^{t} U_{v} (τ) d τ + α \cdot μ_{s} (t) . \end{matrix}$

For a systematic derivation, see Appendix A.

We list all variables and descriptions in the table given next:

Variable	Description
$y_{s} (t), y_{v} (t)$	Continuous-time outputs of the systems corresponding to s and v respectively.
$y_{s} [k], y_{v} [k]$	Discrete-time outputs of the systems corresponding to s and v.
$μ_{s} (t), μ_{v} (t)$	Control actions for the systems s and v respectively in continuous time.
$μ_{s} [k], μ_{v} [k]$	Control actions in the discrete domain for the systems s and v.
$U_{s} (t), U_{v} (t)$	Continuous-time error signals for systems s and v.
$U_{s} [k], U_{v} [k]$	Error signals in the discrete domain for the systems s and v.
T_s	Sampling time, time difference between two consecutive sampling points.
$τ_{s}, τ_{v}$	Time constants for the systems s and v respectively.
$R_{s}^{p}, R_{v}^{p}$	Proportional control gains for the systems s and v.
$R_{s}^{I}, R_{v}^{I}$	Integral control gains for the systems s and v.
$α$	Coupling factor between the two controllers.
$Λ_{s} [k], Λ_{v} [k]$	Setpoint values for the systems s and v in the discrete domain.

Stress network based on Potts model

The Hamiltonian for the q-state Potts model on a graph $G = (V, E)$ is given by: $ℋ = - J \sum_{⟨i, j⟩ \in E} δ (s_{i}, s_{j}) .$ s_i represents the spin at node i taking values in the set $\{1, 2, \dots, q\}$ . J denotes the interaction strength between adjacent nodes, and $δ (s_{i}, s_{j})$ stands for the Kronecker delta function. $⟨i, j⟩$ denotes bonds between nodes i and j respectively.

The partition function Z is given by $Z = \sum_{s} e^{- β ℋ}$ , with $β$ as the inverse temperature that we interpret as the stress factor. The sum of the partition function goes over all possible configurations $\{s\}$ . In the Fortuin-Kasteleyn representation of the Potts model, each bond $i, j$ is present with probability $p = 1 - e^{- β J}$ and absent with probability $1 - p$ . As $q \to 1$ , the Potts model transitions into a bond percolation problem.

For a directed graph $G = (V, E)$ , each (directed) edge in G is occupied (open) with probability p and unoccupied (closed) with probability $1 - p$ , independently of other edges. A node j is reachable from node i if there exists a directed path of open edges leading from i to j. The percolation threshold p_c is the critical probability marking the transition from having no infinite clusters to having one that spans the graph (Fig. 2). We outline our directed bond percolation algorithm for generating the stress network as follows:

Algorithm DirectedBondPercolation

(G (V, E), p, X)

G (V, E) :

Directed graph with vertices V and edges E

// p: Probability for an edge to be open

// X: Set of boundary nodes

//Initialization

ActiveNodes ← X

for each edge

e \in E

e.status

\leftarrow C L O S E D

//Percolation Process

while

A c t i v e N o d e s \neq \emptyset

N e x t A c t i v e N o d e s \leftarrow \emptyset

for each node n in ActiveNodes do

for each outgoing edge e from n do

r \leftarrow r a n d o m (0, 1)

r \leq p

then

e . s t a t u s \leftarrow O P E N

NextActiveNodes ← NextActiveNodes ∪ targetnodeofe

end if

end for

A c t i v e N o d e s \leftarrow N e x t A c t i v e N o d e s

end while

return G

Potential propagation via gap junctions

We use the concept of node potentials in a connected network and the Laplacian to propagate changes in membrane potentials throughout the network. We consider a connected graph $G = (V, E)$ , where V is the set of nodes (cells) with n elements and E is the set of edges (gap junctions) with m elements.

Let $ϕ : V \to ℛ$ be a function that assigns a real number, called potential, to each node in the cell network. The membrane potential of a cell represents its state that changes over time. We define the Laplacian matrix L of the graph $G a s f o l l o w s : L = D - A$ , where D is the diagonal degree matrix and A is the adjacency matrix of G. Specifically, for node i : $D_{i i} = d e g (i)$ is the degree of node i. $A_{i j} = 1$ iff i and j are adjacent, and 0 otherwise.

We classify the nodes of G into two categories: boundary nodes (set B) and non-boundary nodes (set N). Boundary nodes are nodes in the stress network governing the propagation of changing membrane potentials, whereas non-boundary nodes do not participate in the stress network. The Laplacian matrix L can be partitioned based on these categories as: $L = [\begin{matrix} L_{B B} & L_{B N} \\ L_{N B} & L_{N N} \end{matrix}]$

where $L_{B B}$ represents interactions between boundary nodes, $L_{B N}$ represents interactions from boundary to non-boundary nodes, $L_{N B}$ represents interactions from non-boundary to boundary nodes, and $L_{N N}$ represents interactions between non-boundary nodes.

The intracellular dynamics between stress and membrane voltage controller drive changes in membrane potentials resulting in a potential difference $Δ ϕ_{B}$ . The change in potentials in the non-boundary nodes, $Δ ϕ_{N}$ , can be deduced using the Laplacian matrix. Using the equation $L Δ ϕ = 0$ we can partition based on the boundary and non-boundary nodes: $[\begin{matrix} L_{B B} & L_{B N} \\ L_{N B} & L_{N N} \end{matrix}] [\begin{matrix} Δ ϕ_{B} \\ Δ ϕ_{N} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$

From the second set of equations, we can write: $L_{N B} Δ ϕ_{B} + L_{N N} Δ ϕ_{N} = 0$ . Given $Δ ϕ_{B}$ , we can compute $Δ ϕ_{N}$ as: $Δ ϕ_{N} = - L_{N N}^{- 1} L_{N B} Δ ϕ_{B}$ . Note that while the Laplacian matrix L is singular, the submatrix $L_{N N}^{- 1}$ is nonsingular (invertible).

Results

Toy model reveals the stabilizing effect of noise

We first describe an experiment with a toy network to assess the impact of varying levels of stress and voltage coupling among cells on system stability in planarian regeneration. Initially, we generate simulations based on a simplified network of three nodes, representing cells, and we intentionally exclude the role of voltage propagation via gap junctions.

We investigate two core scenarios: one with constant, stable connections between cells, and the other with one connection randomly eliminated at each time step, introducing instability. Further, we segment the investigation into three unique conditions based on different stress and voltage coupling levels, aiming at understanding how these variations affect system disturbances and stability. The broader purpose of this initial analysis is to enhance our comprehension of the intricate cellular mechanisms involved in regeneration.

As discussed in the Models and Methods section, our study is primarily built on dual attributes inherent to each cell: a stress controller and a voltage controller (Fig. 1). A fundamental assumption in our work is that these two mechanisms are internally coupled within each cell, with cellular interactions mediated through the influence of stress among neighboring cells.

We hypothesize that a wide spectrum of stress influences from various cells augments the overall stability of the system, in comparison to a system where stress influence originates from a fewer number of cells. To illustrate this, we use a simple network composed of just three nodes, representing three cells. Our focus is on the dynamics of stress coupling and stress-voltage coupling, excluding voltage propagation via gap junctions for simplicity. This represents the first quantitative model (Fig. 3) implementing the qualitative idea proposed recently,⁵⁷ that stress sharing is a mechanism for coordinating cell behavior into larger-scale anatomical homeostasis.

FIG. 3.

Growth and proliferation processes. (A) Growth direction from tail to head. (B) Growth direction from head to tail. Both (A, B) show the growth front during the regeneration process. Morphogen concentration is low toward tail and high toward head. Newly proliferated cells produce a morphogen level according to the power law (caption Fig. 1). As the morphogen concentration surges in newly proliferated cells, we hypothesize that the membrane voltage setpoints—the predetermined voltage levels that cells strive to maintain across their membranes—exhibit a decrease. This decrease is essential for enabling the cells to enter a state of proliferation and for regulating their developmental processes. This inverse relationship can be attributed to the fact that higher morphogen concentrations typically trigger more intense cell proliferation,^52,53 which, in turn, necessitates a decrease in membrane voltage setpoints. This is due to the fact that lower membrane voltage helps promote cell growth and division, essential processes during proliferation.^54–56 During regeneration processes, we observe a high stress level across the stress network. (C) shows the logic for new proliferated cells. P denotes the stress propagation probability, and p is a random number drawn from a uniform distribution $U (0, 1) .$ In case new proliferated cells belong to the stress network, they actively participate in the membrane voltage propagation through gap junctions (see the Models and Methods section for details). The placement of new cells follows an Eden type of growth model that can be understood as a diffusion-limited aggregation in this context.

We delve into two specific cases. The first scenario involves stable stress connections between cells (represented by the edges in our network) that persist throughout the simulation (see left column in Fig. 4). This implies that all connections between cells are constantly maintained. In contrast, the second scenario introduces a degree of instability, with one connection being randomly eliminated at each time step during the simulation (right column, Fig. 4).

FIG. 4.

Toy network with three nodes. Nodes represent cells, and edges represent intercellular stress coupling between cells. Main columns. Right column: fully connected cell network during simulations. Left column: unstable connections during simulation (we randomly drop one connection each time step). Number of time steps: 100. α denotes the coupling strength between stress and voltage controller. λ denotes the stress coupling between cells. Sub-columns in each panel: stress and voltage levels as a function of time. Coordinates of the state-space; x = stress level, y = voltage level. In all configurations: voltage and stress setpoints drawn from a normal distribution $N (μ = 100, σ = 3)$ . Initial stress deviation from setpoints drawn from a uniform distribution between ±2% and ±10% deviation from corresponding stress setpoints. We set $R_{s}^{p} = R_{v}^{p} = 0.2$ and $R_{s}^{I} = R_{p}^{I} = 0.6 f o r a l l e x p e r i m e n t s .$ (A) $α = 0.3 a n d λ = 0.3$ , (B) $α = 0.05 a n d λ = 0.5$ , and (C) $α = 0.5 a n d λ = 0.05 .$ We observe in all three scenarios more stable dynamics when all three cells remain connected during the full simulation period.

Our research is further diversified into three distinct scenarios based on varying levels of stress and voltage coupling: Scenario A (Fig. 4A), where both stress and voltage couplings are sufficiently high (α = 0.3 and λ = 0.3); Scenario B (Fig. 4B), where stress coupling is high and voltage coupling is low (α = 0.05 and λ = 0.5); and Scenario C (Fig. 4C), where stress coupling is low and voltage coupling is high (α = 0.5 and λ = 0.05). In all scenarios, unstable connections resulted in higher system disturbances.

Our results point to a counterintuitive role of noise in the system: it appears to enhance the overall system stability within the cellular dynamics of planarian regeneration. We interpret this by considering that a higher volume of incoming signals—due to a larger number of coupled cells—increases the probability of certain signals exhibiting destructive interference, which, in turn, stabilizes the system.

Model scenario 1: bisecting a wild-type planarian

We next conducted an experiment focusing on a bisected wildtype planarian. This is a more difficult case than the two-cuts scenario modeled in many planarian papers, because in the case of two cuts it is easy to visualize a morphogen gradient that sets the pattern of the fragment. However, in the case of one cut, the cells on either side of the cut were direct neighbors (in effect, at the exact same positional information level^58,59) and yet have directly opposite anatomical fates (head and tail respectively). This shows that the wound identity cannot be set locally and must arise as the result of long-range communication⁵; those are the dynamics being modeled here.

We created simulations under defined parameters to observe the distinct growth paths, namely tail-to-head and head-to-tail. Our intent was to discern the impact of intracellular stress mechanisms and cell network voltage propagation on membrane potential dynamics during the various phases of planarian regeneration in our model.

The analysis of a wildtype planarian, cut down the middle, is illustrated in Figure 5A, where diverse colors indicate varying membrane potentials in arbitrary units. We investigated two distinct growth trajectories—tail-to-head (top row) and head-to-tail (bottom row), guided by the model outlined in Figure 3. Our simulations considered specific coupling constants for both voltage and stress controllers $(R_{s}^{p} = R_{v}^{p} = 0.3 a n d R_{s}^{I} = R_{p}^{I} = 0.3)$ and a stress network propagation parameter (p = 0.47).

FIG. 5.

Model predictions in the case of a single bisecting cut. (A) Wildtype planarian. Colors indicate membrane potentials in arbitrary units (similar to the pseudocoloring of voltage dye data^14,31,60,61). Cut in the middle of the planarian. Top row. Tail to head growth. Growth process as outlined in Figure 3. Coupling constants for the voltage and stress controllers: $R_{s}^{p} = R_{v}^{p} = 0.3$ and $R_{s}^{I} = R_{p}^{I} = 0.3 .$ Stress network propagation $p = 0.47$ . Bottom row, head to tail growth. Parameter as in top row. We can distinguish three different phases. Phase 1a–1c: high stress activity phase leading to high membrane voltage oscillation. Phase 2: short relaxation phase. We observe less oscillatory activity in already proliferated tissue parts. Phase 3a–3b: increase of activity along generated stress networks during regeneration of higher/lower membrane potential areas. Phase 4: regenerated planarian. We observe that tail and head do not have smooth membrane potentials as generated in the wildtype. (B) Left: regenerated head region. Right: regenerated tail region. Regenerated parts have imprinted membrane voltage structures. These structures are aligned with the generated stress work during regeneration. We hypothesize that these structures store the morphology.

The experimentation was structured to traverse three significant stages of regeneration, all discernible through alterations in membrane potential dynamics influenced by intracellular stress mechanisms and voltage propagation through the cell networks.

The results generated from this experiment were multifaceted. Initially, we observed vigorous oscillations in membrane potentials during phases 1a to 1c, indicative of robust synchronization incited by intracellular stress voltage dynamics. This high-activity phase transitioned into a brief relaxation phase, Phase 2, where oscillatory activity was significantly diminished in areas where tissue proliferation was complete.

The final phase, Phase 3, was characterized by increased synchronization activity along the stress networks, particularly evident toward the latter part of the regeneration process. Interestingly, despite the intense activity, the regeneration process did not perfectly replicate the original planarian, as shown by the “inherited” membrane voltage profile of the regenerated parts, particularly observable in Phase 4.

This inheritance of stress network structure manifested differently in regenerated heads and tails. The average membrane potential in regenerated heads was 5% to 10% higher, whereas regenerated tails exhibited an increased average potential of 2% to 7%, across 10 independent simulations.

Wildtype planarian, cut in the middle, with partial and full gap junction blockage

One of the key observations about long-range control in planaria is the finding that blocking gap junctions after amputation induces permanent changes in the target morphology of planaria.^33,62 In this section of our study, we aimed at elucidating the role of gap junctions, crucial elements facilitating intercellular communication and voltage synchronization, in planarian regeneration.

Specifically, we investigated the effects of partial and total blockage of these junctions on the regeneration process and the resulting membrane voltage structures. To accomplish this, we employed computational simulations with conditions emulating varying degrees of gap junction blockage, and we studied the subsequent regeneration under these modified conditions.

We performed an experiment where we modified conditions to emulate blockage of gap junctions, a crucial element in cellular communication and synchronization. As illustrated in Figure 6, we began with a wildtype planarian that was cut in the middle. We then observed the resulting growth under two distinct conditions: partial gap junction blockage and total gap junction blockage.

FIG. 6.

Wildtype planarian grows into two-headed planarian after exposure to gap junctional blocker. Blockage of gap junctions (such as by octanol^33,62) has been simulated by randomly setting off edges between adjacent cells. We set off two to three connections per cell per timestep. (A) Represents the growth of a two-headed planarian from a mid-section cut wildtype planarian, under conditions of partial gap junction blockage. The parameter settings are identical to those in Figure 5. As a result of the partial blockage, which impedes proper voltage synchronization between neighboring cells, irregular voltage membrane structures are observed in the region near the head. This is attributed to the higher stress levels in the newly proliferated cells, which cause significant disturbances in membrane voltage levels. These disturbances intracellularly provide feedback into sustainable stress setpoint deviations in the new cells. Due to the lack of proper gap-junction based voltage synchronization during the regeneration process, these disturbances are persistent. (B) Illustrates the scenario where complete gap junction blockage leads to the growth of two-headed planarians. Interestingly, the membrane voltage disturbances near the head region are found to be less in comparison to those in the case of partial blockage. In both scenarios—partial and total blockage—similar dynamics to those observed in the experiments without gap junction blockage during phases 1a–1c are noticed. From Phase 2 onward, however, increasing disturbances due to the lack of membrane voltage synchronization during regeneration become apparent. Notably, the patterns resulting from these conditions are found to be stable. This figure suggests a crucial role of gap junctions in the regulation of membrane voltage during planarian regeneration, influencing the emergence of unique morphological outcomes.

The parameters for this experiment were kept consistent with those used in Figure 5. Under partial blockage, our intention was to observe the effects of incomplete voltage synchronization between neighboring cells. With total blockage, we aimed at investigating the growth patterns when the gap junctions were entirely nonfunctional.

Under partial blockage, we noted irregular voltage membrane structures, particularly near the head region of the regenerating planarian. We attribute this to the higher stress levels found in the newly proliferated cells, which caused disturbances in the membrane voltage levels. Interestingly, these disturbances were sustainable and caused deviations in the stress setpoint of the new cells, and due to the partial loss of gap-junction based voltage synchronization, these disturbances were persistent.

On the other hand, when the gap junctions were completely blocked, the regenerated planarian still developed two heads, but the voltage disturbances near the head region were significantly reduced compared with those observed under partial blockage. Irrespective of the degree of blockage, the early stages of regeneration (phases 1a–1c) exhibited similar dynamics to the experiment without gap junction blockage.

However, from Phase 2 onward, we noted increasing disturbances due to the lack of voltage synchronization during regeneration. Remarkably, these observed patterns proved to be stable under both conditions.

Discussion

Many prior efforts have sought to model signaling relevant to planarian regeneration.^{38–40,63–69} Our model adds to this a detailed simulation of gap junctions (bioelectric synapses) and a stress parameter to make explicit the mechanistic dynamics of a homeostatic loop that progressively seeks to reduce error.² Future work will expand the model itself, and apply it to the full range of known planarian functional experiments, to refine it and generate testable predictions.

Additional experiments that need to be simulated include the persistence of the two-head state in subsequent cuts in the absence of octanol,^33,62 the effects of pharmacologically induced voltage change on the number of heads,^14,31 the appearance of a destabilized (randomly choosing one-head vs. two-head in fragments) phenotype,^32,61 and the effects of anxiolytic drugs such as serotonin transporter inhibitors.^70–72

Subsequently, the model can be expanded to include known molecular pathways involved in planarian regeneration downstream of the voltage signals^21,73–77 and other cell behaviors such as proliferation and migration.^78–80

In both our simplified network model and the gap-junction blockage experiment, we discovered that disruption in the connections within the system—either by randomly eliminating connections in the model or by blocking gap junctions in the experiment—resulted in increased system instability. The resilience we initially observed, as well as the subsequent sustainable deviations in membrane voltage distributions, strongly underscores the critical role of these connections in maintaining system stability.

In the case of the gap junction blockage experiment, we noted persistent and sustainable deviations in the membrane voltage levels of the regenerated planarian compared with the normal voltage distribution in the head region of wildtype planarians. This outcome corroborates our findings from the simplified network model where a lack of stable connections resulted in increased disturbances within the system.

Taken together, these results suggest a hypothesis that can be tested in planarian experiments in vivo: stability is highly dependent on the integrity of connections, whether they be stress influences among cells or gap junctions facilitating voltage propagation.

Additional experiments suggested by this model include degrees of reduction of stress; targeting stress perception could be accomplished by means of anxiolytic drugs,^81–85 enabling the planarian model system to shed light on the most interesting and still largely mysterious aspect of morphogenesis: the ability of the system to detect deviations from target morphology and execute self-limiting growth and repair activity.

One limitation of the planarian model system is that it is currently not possible to directly visualize either gap junctional states or stress molecules non-destructively. Being able to visualize this in the living state while planaria regenerate is a key goal for the development of non-invasive optical monitoring techniques, and it will greatly enhance the study of how this versatile, robust model species reliably produces whole bodies from tissue fragments.

In addition to planaria, this model will be adapted to understand size control of organs (e.g., liver) or embryonic/regenerating structures in other model systems, including mammals. In turn, cycles of modeling and empirical testing will reveal sufficient mechanisms by which groups of cells coordinate and make organ-level anatomical decisions.

This knowledge is essential not only for basic evolutionary developmental biology (origin of specific forms and evolvability of repair mechanisms⁸⁶) but also for attempts to exert rational control over large-scale growth and form in biomedical and synthetic bioengineering settings.^87–91

Footnotes

Acknowledgments

We thank Julia Poirier for her help with the manuscript.

Author Disclosure Statement

M.L. is a cofounder of, and the Levin lab receives funding from, a company called Morphoceuticals, which seeks to induce biomedical regeneration by modulating bioelectrical signaling.

Funding Information

M.L. gratefully acknowledges support via Grant 62212 from the John Templeton Foundation (the opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation) and the Guy Foundation Family Trust (103733-00001).

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