# Minimal Developmental Computation: A Causal Network Approach to Understand Morphogenetic Pattern Formation

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## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

## 3. Results

#### 3.1. The Model Learns to Generate the Correct Activity Patterns and Mark Boundaries

#### 3.2. Analysis of Cellular Activity and Structural Patterns

#### 3.2.1. The Model Develops Network Activity and Boundary-Marker Patterns Establishing a Correct Axial Gradient Pattern within the Tissue

#### 3.2.2. The Gap Junctions and Cell Types Also Self-Organize into Patterns Even Though They Were Not Specifically Selected for That Purpose

#### 3.2.3. The Model Successfully Regenerates and Rescales the Pattern despite Not Being Selected for Those Abilities

#### 3.2.4. The Model Generates the Same Qualitative Patterns Regardless of the Initial Network Conditions: Robustness

#### 3.3. Analysis of Intracellular Controller Activity Patterns

#### 3.3.1. Internal Controller Activity Patterns Simultaneously Correlate with Cellular Properties and Network Activity Patterns

#### 3.3.2. Isolated Cells Contain Relevant but Insufficient Information Required to Generate the Network-Level Patterns

#### 3.4. Analysis of Intercellular Causal Network Patterns

#### 3.4.1. Every Cell in the Collective Contains the Full Causal Information about the Network-Level Patterns Explaining the Model’s High Degree of Robustness

#### 3.4.2. The Network Dynamically Integrates into an Organization with Macro-Scale Modules Explaining the Overall Shape of the Functional Patterns

#### 3.4.3. Rescaling the Model Rescales the Causal Networks, Explaining Why the Phenotypic Patterns Rescale

#### 3.4.4. The Overall Structure of the Mean Causal Network Explains the Model’s Ability to Canalize Random Initial States into the Same Patterns

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the modeled biological phenomenon and how machine-learning is used to design the model. (

**a**) During early embryogenesis, a relatively featureless embryo develops distinct axial patterns along with a distinct outer layer known as the epidermis (figure of larva was inspired by [1] (p. 11)). The distinct colors represent the origin of the differentiation of the embryo into distinct morphological features. The cells on the boundary (thick empty circles) represent the epidermis. (

**b**) A model with unknown parameters is trained using machine-learning, that uses gradient-descent-like methods to ‘backpropagate’ the error between the observed and target patterns to the model parameters, to produce the desired pattern.

**Figure 2.**Schematic of the model of axial pattern development. The model comprises of a finite linear chain of cells (a total of 12 cells comprise the model used in this work). We hypothesize that the following elements will be sufficient to implement emergent axial patterning. Every cell has two kinds of signaling networks, one that determines the properties of the cell, namely the cell type and the gap-junction weights, and the other that signals to the cell the extent to which it lies at the boundary. These two factors are expected to act in concert in simultaneously detecting the boundaries and developing a gradient-like phenotypic pattern within the detected boundaries where the pattern would be expected to taper off. The intrinsic controller is a 3 × 3 lattice (totaling 9 nodes), and the boundary controller is a 2 × 3 lattice (totaling 6 nodes), depicted as part of the representative cells $n$ and $(n+1)$ in the middle. As the connections indicate, the anterior and posterior (1st and 3rd) columns of the intrinsic controller influence the anterior and posterior gap-junction weights respectively, and the central column influences the cell’s self-weight. In the same way, the most anterior and posterior (1st and 3rd) columns of the boundary controller signal to the anterior and posterior cells, respectively. Finally, in each cell, the cell state influences all the nodes of either controller, the cell type is influenced by all of the intrinsic controller nodes, and all of the boundary controller nodes are influenced by the cell’s own boundary-marker level.

**Figure 3.**Formal definition of the model. All variables and parameters are continuous in that they can in principle assume any value on the real number line. The parameters (red) are trained and thus fixed during simulations. All variables are scaled by the boundary marker level $b$, except itself, representing its dampening effect (akin to a time constant)—the greater the value of $b$ (the greater the cell tends to be a boundary cell), the more dampened its overall activity, including the patterning state, cell properties and internal controller states, is. Subscripted parameters and variables denote unique projections of the corresponding parameters and variables that can be inferred from the connectivity diagram shown in Figure 2. In cases where the value of a variable is determined by inputs from multiple controllers, such as the gap-junctions and the boundary-markers, the averaging operator $g()$ computes the means of those contributions. The case of gap-junction weight updates exemplifies these concepts. The weight of every gap-junction ${j}_{\left(n-1\right)n}$ is determined by the intrinsic controllers of cells $\left(i-1\right)$ and $i$, specifically by the posterior column in the controller of the anterior cell and the anterior column of the posterior cell. Accordingly, the dimension of ${r}_{g}$ would be $3\times 2\times \left(n-1\right)$, where the dimensions $3\times 2$ represents the three nodes each of the two controller columns. This ${r}_{g}$ would then be partially vectorized, with the two columns contributing to each gap-junction concatenated into one, yielding a matrix with dimensions $6\times \left(n-1\right)$. In the same way, the dimension of ${u}_{g}$ would be $2\times 6$, representing the two gap-junctions each contributed to by one column of the generic cell’s intrinsic controller totaling six nodes (three nodes per column). Thus, the multiplication of ${u}_{g}$ with ${r}_{g}$ yields a matrix of dimensions $2\times \left(n-1\right)$. Finally, the averaging operator $g()$ computes the column-wise mean of the previous matrix resulting in a $1\times \left(n-1\right)$ vector of updates to all the gap-junctions.

**Figure 4.**Training performance. Models trained using machine-learning have low performance errors (blue) compared to the random models (red). The line of blue dots at the bottom comprises the set of top-performing models (72% of the trained models) with similar scores.

**Figure 5.**Patterning and boundary-marking behavior of the best-performing model. When initialized from homogeneous conditions and run for 4000 time-steps, the (

**a**) network activity state pattern converges to a pattern (solid black) that closely matches the target pattern (red), and (

**b**) the normalized boundary marker pattern reaches (solid black) a state where the cells at anterior and posterior poles have the highest levels as desired (target in red). The dashed black lines represent the initial states, and the solid black lines depict the patterns during the last 100 time-steps.

**Figure 6.**Cellular properties of the best-performing model. When initialized from homogeneous conditions and run for 4000 time-steps, the (

**a**) intercellular gap-junction weights and the self-weights and (

**b**) the cell types converge to characteristic shapes (black). Note that the model has eleven gap-junctions connecting the twelve cells in a chain, and every cell has a self-weight and a cell type.

**Figure 7.**Single cells possess an intrinsic controller structure with characteristics resembling PCP. (

**a**) A schematic illustrating the concept of PCP using the analogy of magnetic domains. Ferromagnetic materials contain domains within which the magnetic orientations are aligned (indicated by the arrows). The overall random pattern of orientations (left) could be modified by the application of external forces, such as magnetic fields, or temperature forcing it to assume non-random shapes (right). (

**b**) Likewise, the application of a target gradient-like pattern (Figure 5) enables machine-learning to organize the initially random intrinsic controller weights into PCP-like patterns over phylogenetic timescales. Specifically, the three anterior controller weights (blue) that control the anterior GJ and the three posterior controller weights (red) that control the posterior GJ of a single cell were randomly initialized in the interval [−1, 1] during the training. At the end of training, the anterior and posterior controller weights of the representative model culminated with categorically distinct values, the anterior set positive and the posterior negative, giving the cell a character of polarity. The inset shows a blow-up of a single cell together with its intrinsic controller (Figure 2).

**Figure 8.**Regenerative and rescaling behaviors of the network activity. (

**a**) Regeneration: the model is run for 4000 time-steps following homogeneous conditions, as before, leading to the blue pattern, then all states but that of the middle two cells are zeroed out (green) and run for another 4000 time-steps resulting in the final pattern (red). Even though the blue and red patterns do not exactly coincide they are qualitatively similar to each other. (

**b**) Rescaling: the model is simulated in the same way as Figure 5a, except with 22 cells instead of the original 12 cells. With almost double the number of cells, the model takes about 3.5 times longer (14,000 time-steps) to settle, and moreover it converges (last 100 time-steps shown) to a smoother pattern compared to the 12-cell case.

**Figure 9.**The pattern attractor-space of the activity state. The model converges to patterns (black) that are qualitatively similar to the target pattern (red) when started from a set of 1000 random initial conditions (grey). The initial conditions specifically involved a randomized initial number of ‘active’ cells whose activity states were drawn from the interval [−1, 1] and boundary-marker states from the interval [1, 2] with uniform probabilities. In the case of the ‘non-active’ cells, the activity states were set to 0 and the boundary-marker states were set to 2. The internal controller states were set to 0 in both cases.

**Figure 10.**The intrinsic controller nodes’ activities simultaneously resemble the cell-properties and the network-activity patterns. Each line in the plot represents the asymptotic activity of a particular controller node across the network. That is, each line represents (

**a**) the vector (${r}_{i,1}\left(\tau \right),\dots ,{r}_{i,n}\left(\tau \right))$ for a particular controller node $i\in \left\{1,\dots ,9\right\}$ $\mathrm{at}\tau =4000$ and (

**b**) its cell-normalized version $\left(\widehat{{r}_{i1}},\dots ,\widehat{{r}_{in}}\right)$ where $\widehat{{r}_{ij}}=\frac{\left({r}_{ij}-\underset{1\le j\le n}{\mathrm{min}}{r}_{ij}\right)}{\left(\underset{1\le j\le n}{\mathrm{max}}{r}_{ij}-\underset{1\le j\le n}{\mathrm{min}}{r}_{ij}\right)}$.

**Figure 11.**Single cells contain relevant but insufficient dynamical information about the network-level pattern. When a single cell is isolated and its external input, namely the cell activity state, is clamped and simulated for 2000 time-steps (half the time required by the network to converge), then (

**a**) the internal controller nodes converge to states that clearly discriminate between the various clamped inputs. However, (

**b**) the cell properties, namely the cell type, the two gap-junction weights, the self-weights converge to the same values in the respective categories regardless of the clamped input. In both cases, the states were centered at their respective mean values.

**Figure 12.**Individual nodes in the intrinsic controller network of every cell possess information about the network-level activity pattern that they control. Each line in the plot represents the normalized causal influence exerted by the initial state $\left(t=0\right)$ of a single internal controller node in a specific cell over the asymptotic $\left(\tau =3500\right)$ activity states of the (influenced) cells, that is, it is the normalized vector $\left(\frac{\widehat{\partial {s}_{1}\left(3500\right)}}{\partial {r}_{j,k}\left(0\right)},\dots ,\frac{\widehat{\partial {s}_{n}\left(3500\right)}}{\partial {r}_{j,k}\left(0\right)}\right)\mathrm{for}$ a specific controller node $j\in \left\{1,\dots ,9\right\}$ in the influencing cell $k\in \left\{1,\dots ,n\right\}$ where, $\widehat{\frac{\partial {s}_{i}\left(\tau \right)}{\partial {r}_{j,k}\left(0\right)}}=\frac{\frac{\partial {s}_{i}\left(\tau \right)}{\partial {r}_{j,k}\left(0\right)}-\underset{1\le i\le n}{\mathrm{min}}\frac{\partial {s}_{i}\left(\tau \right)}{\partial {r}_{j,k}\left(0\right)}}{\underset{1\le i\le n}{\mathrm{max}}\frac{\partial {s}_{i}\left(\tau \right)}{\partial {r}_{j,k}\left(0\right)}-\underset{1\le i\le n}{\mathrm{min}}\frac{\partial {s}_{i}\left(\tau \right)}{\partial {r}_{j,k}\left(0\right)}}$.

**Figure 13.**Causal network integration behind the network activity pattern developed under homogeneous initial conditions. An arrow from cell $j$ to cell $k$ represents the causal influence $\partial {s}_{k}\left(\tau \right)/\partial {r}_{i,j}\left(0\right)$ where $\exists i:\partial {s}_{k}\left(\tau \right)/\partial {r}_{i,j}\left(0\right)$ is a statistical outlier in the set $\left\{\frac{\partial {s}_{k}\left(\tau \right)}{\partial {r}_{1,j}\left(0\right)},\dots ,\frac{\partial {s}_{k}\left(\tau \right)}{\partial {r}_{9,j}\left(0\right)}\right\}$. Blue links represent positive influence and red links represent negative influence. Multiple arrows originating from a cell may be associated with distinct intrinsic controller nodes of the originating cell.

**Figure 14.**Rescaling the model (double the number of cells) rescales the corresponding causal network attractor underlying the network-activity pattern. The (

**a**) causal network attractor and (

**b**) its schematized version following rescaling of the model and simulating it with homogeneous initial conditions. The causal network attractor following regeneration is not shown, as it looks identical to the original (Figure 13).

**Figure 15.**The mean causal network attractors associated with the network activity patterning. The thickness of the edges represents the frequency with which they appear in the set of attractors. The initial conditions that were used here are the same as those described in Figure 9.

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Manicka, S.; Levin, M.
Minimal Developmental Computation: A Causal Network Approach to Understand Morphogenetic Pattern Formation. *Entropy* **2022**, *24*, 107.
https://doi.org/10.3390/e24010107

**AMA Style**

Manicka S, Levin M.
Minimal Developmental Computation: A Causal Network Approach to Understand Morphogenetic Pattern Formation. *Entropy*. 2022; 24(1):107.
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**Chicago/Turabian Style**

Manicka, Santosh, and Michael Levin.
2022. "Minimal Developmental Computation: A Causal Network Approach to Understand Morphogenetic Pattern Formation" *Entropy* 24, no. 1: 107.
https://doi.org/10.3390/e24010107