On Growth and Form, and Function: Reusable Regulatory Handles Control Phenotypic Variation

Benedikt Hartl; Milton L. Montero; Marcello Barylli; Sebastian Risi; Michael Levin

cs.NESep 24, 2026

On Growth and Form, and Function: Reusable Regulatory Handles Control Phenotypic Variation

Authors: Benedikt Hartl, Milton L. Montero, Marcello Barylli, Sebastian Risi, Michael Levin

Organizations: Allen Discovery Center at Tufts University, Medford, MA, USA · IT University of Copenhagen, Denmark · Sakana AI, Japan · Wyss Institute for Biologically Inspired Engineering at Harvard University, Boston, MA, USA

Abstract

How phenotypic transformations are implemented by changes in underlying regulatory dynamics remains a central question in developmental biology. Inspired by D'Arcy Thompson's 1917 "On Growth and Form", we ask whether coherent large-scale transformations of morphology can be encoded as low-dimensional modulations of a self-organizing developmental system. We use neural cellular automata (NCAs) as bio-inspired models of distributed development, in which a shared local regulatory network grows target morphologies from a single cell. We apply low-rank adaptation (LoRA) to pretrained NCAs, representing each adapted developmental program as a low-rank modulation of a fixed regulatory scaffold. Horizontal and vertical scaling of a fully grown 2D emoji phenotype can each be implemented by rank-one adaptations. Their linear combinations parametrically control phenotype size, generalize beyond the training distribution, and compose with target-specific adapters. Strikingly, adaptations learned for one phenotype transfer zero-shot across structurally and semantically diverse phenotypes sharing the same reference scaffold, while largely preserving internal features. This suggests reusable system-level hyper-directions of scale rather than morphology-specific transformations. From approximately 25,000 independently trained phenotype-specific NCA adapters with a shared scaffold, we further identify latent low-dimensional directions that functionally control phenotypic variation including scaling, style, and symmetrical fission. Together, our results provide a computational realization of D'Arcy Thompson's remarkable grid transformations in a 2D NCA---a minimal cybernetic tissue in which variations of fully grown emoji phenotypes can be encoded, combined, and controlled through low-dimensional directions in regulatory weight space.

Figures & tables

Figure 1: Modulations of self-orchestrated morphogenesis canalize phenotypic transformations and construct a semantic weight space of the regulatory machinery. (A) D’Arcy Thompson’s Cartesian grid transformations relate phenotypic shapes, here using the example of a porcupinefish being transformed into a sunfish, reproduced from Ref. 86 ; public domain. (B) Schematic flow diagram of how modulations at the regulatory level of self-orchestrated morphogenesis canalize phenotypic transformations. Multicellular morphogenesis can be modeled by neural cellular automata (NCAs) which comprise cellular agents whose numerical states are updated by a shared artificial neural network (ANN) that guides cellular perception and cell state updates within the local context of a multicellular grid. While cell states in NCAs resemble physiological states of biological cells, their ANNs operationalize gene-regulatory networks (GRNs) responsible for local intercellular decision-making to achieve collective system-level outcomes. Low-dimensional modulations of the regulatory parameters—ANN parameters, in our case—induce adapted cellular behavior at the microlevel, which can be canalized into effective system-level transformations. (C) Training NCAs on diverse emoji targets reveals a semantically structured and controllable weight space. In this work, we fine-tuned $\approx 25{,}000$ shared-scaffold NCAs via low-rank adaptation (LoRA). The LoRA weights organize into a semantic weight space, as visualized by the UMAP dimensional-reduction scatter plot that is colored by emoji group (see legend). Dominant principal components in the LoRA weight space control global target features such as phenotypic size; linearly translating the ANN of a baseline NCA parameters along those principal directions yields progressive high-level shape transformations while retaining phenotype-specific features.

Figure 2: Schematic illustration of mapping composite phenotypic transformations to mechanistic modulations in an NCA’s regulatory network via LoRA. We perform phenotypic variations $\boldsymbol{Y}\mapsto\boldsymbol{Y}_{xy}=\mathcal{T}_{xy}(\boldsymbol{s})\boldsymbol{Y}$ via parametric transformations $\mathcal{T}_{xy}\left(\boldsymbol{s}=\{s_{x},s_{y}\}\right)$ and ask which LoRA parameter modulations $\theta_{xy}(\boldsymbol{s})=\theta+\Delta\theta_{xy}(\boldsymbol{s})$ give rise to modulated system-level dynamics $\tilde{\boldsymbol{X}}^{t}_{xy}(\boldsymbol{s})$ of a baseline NCA that resemble the transformed phenotype $\tilde{\boldsymbol{X}}^{T}_{xy}(\boldsymbol{s})\approx\mathcal{T}_{xy}(\boldsymbol{s})\boldsymbol{Y}$ after $T$ development steps.

Figure 3: Contrasting (A) target scaling transformation to (B) phenotypic outcomes of modulated NCA dynamics and (C) generalization experiments comparing out-of-distribution (ood) NCA modulations to expected target transformations. (A): $xy$ -scaling transformations $\boldsymbol{Y}_{xy}=T_{y}(s_{y})T_{x}(s_{x})\boldsymbol{Y}$ of a base fish emoji $\boldsymbol{Y}$ (magenta box) from $(s_{0}\times s_{0})=(37x37)$ pixels to $25$ combinations of $(sx\times s_{y})$ pixels with $s_{x},s_{y}\in\{25,31,37,43,49\}$ . (B): Exemplary outcomes $\tilde{\boldsymbol{X}}_{xy}^{T_{D}}$ of the developmental process of $(s_{x},s_{y})$ -parametrically modulated NCAs, see Equation 6 , after $t=T_{D}$ steps. (C): Heatmap of the mean perceptual loss 102 (LPIPS, see Section B.1 ) comparing the phenotypic outcomes of ood-modulated NCAs to the expected but withheld targets across a fine grid of scale transformations $s_{x}^{\prime},s_{y}^{\prime}\in[15,57]$ ; we used $16$ independent stochastic NCA rollouts per scaling and refer to Figure 6 for more details.

Figure 4: Low-rank regulatory adaptations learned to rescale one emoji transfer zero-shot to diverse phenotypes. LoRAs trained on a discrete set of $xy$ -scaling transformations for a single phenotype $\boldsymbol{Y}$ —with $s_{x},s_{y}\in\{25,31,37,43,49\}$ , see Figure 3 —not only generalize zero-shot to out-of-distribution scaling $s_{x}^{\prime},s_{y}^{\prime}$ , but also transfer across phenotypes, i.e., to NCAs with shared reference scaffold $\boldsymbol{W}_{0}$ but fine-tuned on a different target $\boldsymbol{Y}_{\phi}$ . Column (A) shows developmental outcomes for three different targets across selected sizes $s_{x}^{\prime},s_{y}^{\prime}\in\{17,25,31,37,43,49,55\}$ —from top to bottom: a similar fish phenotype to the original blue-fish emoji in Figure 3 , but with different color-composition and internal features; completely different green lizard phenotype; and a red fire extinguisher. Column (B) depicts the mean perceptual loss 102 (LPIPS, see Section B.1 ) per emoji (row) across a fine grid of scale transformations $s_{x}^{\prime},s_{y}^{\prime}\in[15,57]$ ; we used $16$ independent stochastic NCA rollouts per ( $\phi$ , $s_{x}^{\prime}$ , $s_{y}^{\prime}$ ) tuple; refer to Figure 6 for the corresponding perceptual loss STD and MSE loss; the latter is, however, mostly correlated with the overall phenotypic size rather than actual pixel-wise or semantic accuracy of the modulated developmental outcomes. Especially from the perceptual loss (B) we learn that all of the tested target phenotypes $\phi$ can be scaled via the same hyper-directions $\boldsymbol{\Delta}_{x,y}$ , even beyond training conditions; the white circles in (B) emphasize training conditions, where training has only been conducted on the blue fish target shown in Figure 3 . This strongly suggests that these LoRAs are universal directions of scale for NCAs with shared scaffold.

Figure 5: Weight-Space Principal Component Analysis (PCA) reveals Semantic Hyper-Directions across NCA Growth-Dynamics. (A) Projection of full adapter weight modulations, $\Delta\boldsymbol{W}_{\phi}=\boldsymbol{A}_{\phi}\boldsymbol{B}_{\phi}$ , onto leading six principal components, $PC_{0\dotsc 5}$ , colored by the relative horizontal ( $x$ ) extent of the corresponding target emojis $\boldsymbol{Y}_{\phi}$ . The first two principal components, $PC_{0}$ and $PC_{1}$ , strongly correlate with horizontal ( $x$ ) and vertical ( $y$ ) phenotypic extent, respectively ( $r(PC_{0},x)=-0.71$ and $r(PC_{1},y)=-0.73$ , measured from the targets’ bounding boxes). (B) In the top (second-to-top) panel, the growth-dynamics of a baseline cowboy-hat NCA is compressed into $x$ ( $y$ ) direction by adding $\beta\,\Delta\boldsymbol{W}_{PC_{0}}$ ( $\beta\,\Delta\boldsymbol{W}_{PC_{1}}$ ) to its phenotype-specific weight modulation $\Delta\boldsymbol{W}_{\phi}$ , where $\Delta\boldsymbol{W}_{PC_{i}}$ denotes the corresponding principal direction mapped from standardized PCA coordinates back to the original weight coordinates. In the second-to-bottom (bottom) panel, we shrink (stretch) the growth-dynamics of the same NCA into $x$ and $y$ directions simultaneously by adding (subtracting) $\beta\,(\Delta\boldsymbol{W}_{PC_{0}}+\Delta\boldsymbol{W}_{PC_{1}})$ to (from) baseline weights. Unlike the universal scaling directions $\boldsymbol{\Delta}_{x,y}$ , the principal components $PC_{0,1}$ also affect the brightness of the modulated NCAs’ outcomes. (C) UMAP projection 57 of the leading 50 PCs from (A); coloring by vendor, and especially the separation of the “OpenMoji” style (salmon) from the remaining vendors reveals that the weight-space captures style-specific information. (D) The mean weight-space translation vector $\Delta\boldsymbol{W}_{\mathrm{open}}=\langle\Delta\boldsymbol{W}_{\phi}\rangle_{\mathrm{open}}-\langle\Delta\boldsymbol{W}_{\phi}\rangle_{\mathrm{non-open}}$ from NCAs of any vendor (all but salmon in C) to OpenMoji NCAs (salmon in C) yields a style-transfer hyper-direction for OpenMoji-likeness. Four selected NCA experiments (from top to bottom) demonstrate that baseline features (left panels) are preserved by the OpenMoji style-transfer, but an OpenMoji-typical blackish outline (cf. center panels) is “added” to the NCAs’ developmental dynamics (right panels). (E) A “fission”-hyper-direction $\Delta\boldsymbol{W}_{\mathrm{F}}$ has been identified by subtracting the centroid of the remaining NCAs from that of the symmetrically split targets (see top-row examples), $\Delta\boldsymbol{W}_{\mathrm{F}}=\langle\Delta\boldsymbol{W}_{\phi}\rangle_{\mathrm{split}}-\langle\Delta\boldsymbol{W}_{\phi}\rangle_{\mathrm{non-split}}$ , which can be used to parametrically split (fuse) baseline growth-dynamics along the vertical axis by adding (subtracting) $\beta\Delta\boldsymbol{W}_{\mathrm{F}}$ to (from) any baseline weights $W_{\phi}$ .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Low-rank regulatory adaptations induce scale transformations that generalize beyond training conditions and across emoji phenotypes. LoRAs trained on a discrete set of $xy$ -scaling transformations for a single phenotype $\boldsymbol{Y}$ —with $s_{x},s_{y}\in\{25,31,37,43,49\}$ , see Figure 3 —not only generalize zero-shot to out-of-distribution scaling $s_{x}^{\prime},s_{y}^{\prime}$ , but also transfer across phenotypes, i.e., to NCAs with shared reference scaffold $\boldsymbol{W}_{0}$ but fine-tuned on a different target $\boldsymbol{Y}_{\phi}$ . Column (A) shows developmental outcomes for four different targets across selected sizes $s_{x}^{\prime},s_{y}^{\prime}\in\{17,25,31,37,43,49,55\}$ —from top to bottom: the original fish emoji $k$ ; a similar fish phenotype, but with different color-composition, internal features; completely different green lizard phenotype; and a red fire extinguisher. Column (B) and (C) depict the mean and STD of the perceptual loss 102 (LPIPS, see Section B.1 ) per emoji (row) across a fine grid of size transformations $s_{x}^{\prime},s_{y}^{\prime}\in[15,57]$ ; we used $16$ independent stochastic NCA rollouts per (target, $s_{x}^{\prime}$ , $s_{y}^{\prime}$ ) tuple. Column (D) depicts the pixel-wise mean-square error loss between transformed targets and developmental outcomes of the modulated NCAs, which is, however, mostly correlated with the overall phenotypic size rather than actual pixel-wise or semantic accuracy of the modulated developmental outcomes. Especially from the perceptual loss (B) we learn that all of the tested target phenotypes $\phi$ can be scaled via the same hyper-directions $\boldsymbol{\Delta}_{x,y}$ , even beyond training conditions; the white circles in (B-D) emphasize training conditions, where training has only been conducted on the blue fish target. This strongly suggests that these LoRAs are universal directions of scale for NCAs with shared scaffold.

Figure 7: Principal Component Analysis (PCA) of the LoRA-NCA Dataset from Section 3.4 . Top/middle/bottom rows, from left to right: Variance explained/PCA-projection/UMAP-projection of (A) raw LoRAs, (B) SVD-canonicalized LoRAs, and (C) full weight-space. Prior to PCA, parameters are flattened, concatenated, and standardized to zero-mean and unit-variance per dimension.

Figure 8: Outcomes of NCAs constructed with Truncated Numbers of Leading Principal Components (PCs) of the LoRA-NCA Dataset from Section 3.4 . (A) Mean square error (MSE) of NCA outcomes whose weights are truncated linear combinations of the leading PCs (horizontal axis) of canonicalized LoRA factors (cf. Figure 7 B) for individual (grey) and averaged across (blue) multiple randomly chosen target morphologies, respectively normalized to MSE with one leading PC. Thousands of PCs are necessary to reach good accuracy; we report similar results for PCs of raw LoRAs, cf. Figure 7 A. (B) Same as (A) but for the full weight space (cf. Figure 7 C). Accuracy improves with tens and hundreds of PCs already, but thousands of PCs are necessary to get to high-fidelity reconstructions of the full NCA dynamics. (C) Randomly chosen example developmental outcomes (vertical axis) of NCAs constructed from leading PCs (horizontal axis) of canonicalized LoRA factors, cf. (A). (D) Same as (C) but for the full weight space, cf. (B).

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