Digital cousins: Simultaneous optimization of one model for BMP signaling in distant relatives reveals essential core

Abstract

Spatially distributed, nonuniform morphogen gradients play a crucial role in tissue organization during development across the animal kingdom. The Bone Morphogenetic Protein (BMP) pathway, a well-studied morphogen involved in dorsal-ventral (D-V) axis patterning, has been extensively studied in zebrafish, Drosophila, and other organisms. Given that this pathway is highly conserved in both form and function, we sought to determine whether a core mathematical model that constrained topology and biophysical parameters could fully reproduce the observed dynamics of gradient formation in both Drosophila and zebrafish through changes in expression only. We used multi-objective optimization to simultaneously fit a single core model to Drosophila and zebrafish data and conditions. By exploring a single model with varied parameters, we identified both the homology and diversification of the BMP pathway. We find that variation in a small subset of parameters—particularly diffusion-related rates—can reconcile the experimentally measured BMP gradients in both species under wild-type conditions, whereas fitting both WT and mutant conditions requires additional species-specific regulatory extensions beyond the core model. This approach, involving simulation and multispecies optimization, provides a systematic method to explore the minimal parametric variations needed to account for interspecies differences in a developmental pathway. Rather than making predictive claims, our finding offers a framework for improving the interpretability and translational relevance of cross-species models.

Graphical Abstract

1. Introduction

Structures and body plans in organisms across the animal kingdom are specified by spatially distributed nonuniform gradients of molecules called morphogens. These gradients guide the development of patterns from a single-cell structure into a well-organized, patterned organism with diverse structures [1], [2], [3], [4], [5], [6]. The precise positioning of patterned features of an organism is crucial, as they emerge from an initially homogeneous group of cells that differentiate in response to morphogen cues. An extensively studied morphogen is the Bone Morphogenetic Protein (BMP), a member of the TGF-β superfamily, which plays a key role in both vertebrates and invertebrates within the animal kingdom. Of current interest is comprehending how BMPs orchestrate the dorsal-ventral (D-V) axis in the early embryos of zebrafish (a vertebrate) and Drosophila (an invertebrate). Despite their evolutionary divergence, D-V patterning is established by a common pathway in these diverse species through the interaction of BMP ligands (Decapentaplegic (Dpp)/BMP2/4) with extracellular regulators, especially the antagonistic interaction between regulators of the BMP pathway. Fig. 1 illustrates the BMP gradient in patterning the D-V axis in Drosophila and zebrafish embryos. The pathway is initiated by a BMP ligand binding and forming a complex with transmembrane serine-threonine kinase (Type I and Type II) receptors [7], [8]. The binding of BMP ligands to receptors is tightly regulated by extracellular modulators, including antagonists such as Sog/Chordin (Chd) and/or activators such as Tolloid (Tld) (Fig. 1I). As shown in Fig. 1I, upon BMP ligand binding, the Type II receptors phosphorylate the Type I receptors, which in turn phosphorylate members of the R-Smad [9], [10], including Smad 1, Smad 5, and Smad 8. Phosphorylated R-Smads then bind to Smad 4 to form complexes that translocate to the nucleus, where they accumulate and regulate gene expression [11], [12], [13], [14], [15], [16], [17], [18], which leads to precise pattern formation. A simplified representation of the key BMP regulators involved in D-V axis patterning in arthropods and vertebrates is illustrated in Fig. 1I.

Fig. 1.

(A) and (B) depict the dorsal-ventral patterning in Drosophila and zebrafish, respectively. (Copyright: National Museum of Natural History). (C) and (D) display the signaling gradient profiles and expression domains of the Dpp-patterned Drosophila embryo and the BMP-patterned zebrafish embryo, respectively. Panels (E) and (F) show the corresponding cross-sectional views. The arrows in (C) and (D) indicate the directions along which the cross-sections shown in (E) and (F) were taken. (G) presents an experimental average profile of P-Smad domains in the Drosophila wild-type embryo (blue) and Sog loss-of-function embryo (red) (275 μm; transverse section). Similarly, (H) shows the experimental average profile of P-Smad domains in the Zebrafish wild-type embryo (blue) and Chordin loss-of-function embryo (red) (700 μm; transverse section). (I) Schematic of BMP-Smad pathway and signaling regulation by extracellular modulators in zebrafish and Drosophila. BMP2/7 ligands in zebrafish (blue) and Dpp/Scw ligands in Drosophila (green) are regulated by multiple antagonists and cofactors, including Chordin/Sog, Noggin, Sizzled, Tolloid, and CV-2/BMPER. Red inhibitory lines indicate repression, while arrows indicate activation or facilitation. Color coding highlights species context: blue labels indicate zebrafish-specific components, green labels indicate Drosophila-specific components, and black labels denote regulatory factors shared between both systems. Created in BioRender. Li, L. (2025) https://BioRender.com/w21d345.

The classical BMP signaling pathway (Fig. 1I) demonstrates a conservation of the pathway at the cellular level, even though distinct patterns are observed between Drosophila and zebrafish at the organism level (Figure G and H). Understanding the mechanisms of BMP regulation will help distinguish the conserved and divergent components of the signaling pathway between zebrafish and Drosophila, providing insights into how its dysregulation contributes to developmental abnormalities [19], [20]. Homology between zebrafish and Drosophila is evident through the major regulators and their functions. In zebrafish, BMP ligands, BMP2/7, are homologous to Drosophila Dpp/Scw, while Chd, a homolog of Drosophila Sog, acts as an inhibitor of the signal. Even with this level of conservation, when Chd was introduced to Drosophila, Chd failed to fully rescue Drosophila sog^−/− embryos to levels of WT BMP [21], suggesting an evolutionary divergence in the mechanism. Drosophila Sog has been thought to play a paradoxical role as both an inhibitor and a promoter of BMP [22], [23], [24] signaling, while Chd, in zebrafish, does not exhibit this dual function. In addition to this biochemical difference between Drosophila and zebrafish, there are also substantial differences in the spatial and temporal scales [25], [26]. Drosophila has a smaller embryo, less than half the size of the zebrafish embryo. In terms of time scale, the Drosophila Dpp gradient is established to pattern the DV axis in approximately 1 h after cellularization [16], [27], while the zebrafish BMP gradient develops gradually in about 2–3 h [17], [28]. The establishment of the BMP gradient depends on the balance of all these factors: the biochemical reactions, embryo geometry, and the length and time scales of the systems.

Multi-objective optimization (MOO) is an approach that allows model parameter optimization using multiple datasets by defining the different datasets as different objectives [29], [30] and will enable researchers to consider multiple conflicting objectives simultaneously. Different datasets may vary in accuracy and importance on different scales, which are often unknown; therefore, they cannot simply be aggregated into a single objective without proper weights. Unlike single objective optimization, MOO provides a set of compromise solutions, known as the Pareto front, that reveals all trade-offs between objectives. Although MOO provides a desirable approach for integrating all experimental data without prior knowledge of precision, its high computational cost has limited its use in biological systems. Areas that have applied MOO in biology are limited to parameter estimation [31], [32], [33], [34], [35], [36], [37], optimal control [38], [39], [40], [41] and experimental design [42], [43], [44]. Most of these approaches use simulated data instead of real experimental data to demonstrate the methodology.

In this work, we use a mathematical model analysis and multi-objective optimization strategy to study and understand the minimum requirements on the biophysical parameters that balance the known geometry, length, and time scale differences between Drosophila and zebrafish to produce observed BMP gradient data. We investigated how a single model could fit different systems by varying the model parameters to determine the homology (conservation) and also identify the diversification of the BMP pathway in both systems. Mathematical models of BMP signaling have provided some valuable insights into the gradient formation of BMP and how it might be regulated by modulators [4], [45], [46], [47], [48], [49], [50]. These models describe how the morphogen concentrations change over time and space through diffusion and chemical reactions such as synthesis, degradation, association, and dissociation with other factors (such as antagonists). We use a simple core model that captures the classical interactions of the homologous BMP regulators in both Drosophila and zebrafish. This model relies on the balance between diffusion, reaction, and degradation of a locally produced signal [1], [46], [51]. The signal can be realized locally by protein synthesis and could be degraded through proteolysis or receptor-mediated endocytosis, while diffusion is a result of ligand internalization and recycling. The interaction between these processes generates different variations of the model, such as free diffusion (diffusion-decay) models [46], [51], [52], [53], [54], hindered diffusion (reaction-diffusion) models [46], [55], [56], [57], [58], [59], and facilitated diffusion (shuttling) models [4], [23], [45], [47], [48], [58], [60], [61]. It has been shown that other regulatory processes are essential to produce the observed BMP signal in both Drosophila and zebrafish. For Drosophila, positive feedback in response to pMad signaling is needed to further concentrate the surface localized Dpp/Scw at the dorsal midline [27]. In zebrafish, other inhibitors, Noggin (Nog) and Follistatin, in addition to Chd, are essential for the establishment of some ventral cell fates in the early embryo [62]. Our work aims to investigate whether a single mechanism that is homologous exists between Drosophila and zebrafish to produce observed phenotypes.

Pareto fronts provide valuable insights into the relationships between objectives in a system, models or datasets. The shape of the Pareto front reveals how objectives interact: a convex shape illustrates a lot of indistinctness in the objectives, with a steeper shape indicating the potential for a single compromise solution that adequately satisfies both objectives [63]. In contrast, a linear Pareto front suggests that the model is not capable of meeting both objectives simultaneously, revealing a degree of both indistinctiveness and distinctiveness of the objectives. A concave shape, on the other hand, suggests that the objectives are completely distinct and cannot be optimized simultaneously. Pareto fronts can also be compared for dominance using the hypervolume quality indicator (also known as S-metric or Lebesgue measure) [64], [65], [66]. While these tools are available to compare Pareto fronts, they have not been used in biology because of their computational expense. In addition, little work has been done to address uncertainty associated with frontiers in multi-objective optimization, as most uncertainty propagation methods have been developed for single-objective problems. Extending these methods to multi-objective optimization remains computationally expensive and unexplored. In this work, we introduce a novel concept of propagating uncertainty to a Pareto front to account for data uncertainty. Additionally, we extend the concept of Pareto optimality by introducing the Utopian front, a new frontier that relaxes constraints to identify the minimum set of design variables required to be different to achieve a solution that converges to the utopian point that occurs with no constraints and complete independence of design variables in model optimization. This extension of Pareto optimality provides a powerful tool to analyze and identify limitations in complex systems with competing objectives.

In a recent study, we investigated how the BMP/Smad pathway balances multiple systems-level behaviors—such as response speed, noise filtering, and sensitivity—across various cellular contexts with the Pareto front analysis [67]. Despite the pathway’s core connectivity remaining largely unchanged, our findings showed that non-conserved parameters (NCPs), including protein concentrations in the Smad pathway and especially nuclear phosphatase, can be tuned to optimize different performance objectives. However, the Smad pathway alone cannot simultaneously maximize all objectives. In this study, we extend this multi-objective optimization perspective to an organismal scale—contrasting Drosophila and zebrafish—and demonstrate how key extracellular parameters also shape BMP signaling outcomes across species.

2. Methods

2.1. Data collection and calibration

We aggregated measurements for BMP signaling from literature sources and previously published work in various formats for both Drosophila and zebrafish. The primary data of interest included the intensities of the BMP signal transducer, pMad in Drosophila, and pSmad in zebrafish, measured across the D-V axis of their respective embryos. A summary of the data is provided in Table 1. The available data were (1) quantitative (pMad/pSmad intensities), (2) semi-quantitative (images that reflect the spatial distribution of the signaling molecules), and (3) qualitative (descriptive interpretations of the observed phenotype of wild-type and mutant embryos). To ensure consistency, all collected data were calibrated against internally processed experimental data and scaled based on intensity and geometry for both systems [68], [69], [70] for model fitting (All data available in the supplement). These processed data sets were then used to compare with simulation outputs for quantitative model fitting and validation. Drosophila embryo data were collected from dorsal-oriented embryo images. A rectangular region, measuring 10 % the width of the embryo, was then cut in the middle of the A-P axis and averaged to get a one-dimensional pMad distribution along the DV axis. Since the dorsal view image only shows half the embryo, the BMP gradient is positioned from the dorsal to the ventral lateral regions of the embryo. The position of the high BMP signal is labeled position x = 0, dorsal midline for the Drosophila embryo, and the ventral midline for zebrafish.

Table 1.

Data extracted from published literature.

Reference	Embryo genotype	Organism
Umulis et al. (2010)	WT sog+/−	Drosophila
Peluso et al. (2011)	WT sog−/−; 2x sog-i sog+/− sog−/−	Drosophila
Zinski et al. (2017)	WT chd LOF nog LOF	zebrafish

We built a single core model to determine parameters wherein one model could fit both systems, and if not, identify the minimum number of changes in the parameters needed to fit both systems fully. We used the simplest core model shared between Drosophila and zebrafish to generate a BMP profile to fit against the experimental data. The model considered here is a simplified version of our previously published BMP models [71]. This mechanism relies on the diffusion and degradation of a locally produced signal [1], [51], [72], [73]. To form an appropriate simplified model for both Drosophila and zebrafish systems, we applied a set of assumptions: (i) reactions follow first and second-order kinetics, (ii) the system is represented in one dimension, and (iii) BMP heterodimer formation occurs prior to secretion via a covalent bond between monomeric subunits, allowing us to model only a single molecule. For the analysis, we focused on key variables that control BMP inhibitor expression strength and the boundaries of the Chordin/Sog expression and in situ expression data.

Based on these assumptions, we formulated a system of partial differential equations (PDEs) to describe the interactions of regulatory factors. (1), (2), (3) represent a one-dimensional line of cells oriented along the dorsal-ventral axis, where long-range diffusion is approximated using the finite difference method, shown in Eq. 4. We applied no-flux boundary conditions to both of the systems, and all factors were initially set to a null concentration (zero value). The complete set of equations for the simple model is as follows:

$$\frac{\partial [B]}{\partial t}=D_B{\nabla }^2\left[B\right]+{\mathrm{\varnothing }}_B\left(x\right)-k_1\left[B\right]\bullet \left[I\right]+k_{-1}\left[\mathit{BI}\right]-{\delta }_B\left[B\right]+{\lambda }_{\mathit{BI}}\mathit{Tld}\bullet [\mathit{BI}]$$ (1)

$$\frac{\partial [I]}{\partial t}=D_I{\nabla }^2\left[I\right]+{\mathrm{\varnothing }}_I\left(x\right)-k_1\left[B\right]\bullet \left[I\right]+k_{-1}\left[\mathit{BI}\right]-{\delta }_I\left[I\right]-{\lambda }_I\mathit{Tld}\bullet [I]$$ (2)

$$\frac{\partial [\mathit{BI}]}{\partial t}=D_{\mathit{BI}}{\nabla }^2\left[\mathit{BI}\right]+k_1\left[B\right]\bullet \left[I\right]-k_{-1}\left[\mathit{BI}\right]-{\delta }_{\mathit{BI}}\left[\mathit{BI}\right]-{\lambda }_{\mathit{BI}}\mathit{Tld}\bullet [\mathit{BI}]$$ (3)

where [B], [I], and [BI] represent the concentrations of BMP, the inhibitor, and the BMP-inhibitor complex, respectively. The model is parameterized by D_j, the diffusion rate, δ_j, the decay rate, and Φ_j (x), the production rate of protein j, where j = {[B], [I], [BI]}. BMP binds to the inhibitor with a rate of k₁ and unbinds with a rate of k₋₁, Tld cleaves BMP bound to the inhibitor with a rate of λ_BI and cleaves free inhibitor with a rate of λ_I. The inhibitor, [I], is Sog in Drosophila and Chd in zebrafish.

$${\nabla }^2\left[B\right]\approx \frac{{\left[B\right]}_{i-1}-2{\left[B\right]}_i+{\left[B\right]}_{i+1}}{△x^2}$$ (4)

The solution depends on the balance of time scales, diffusion, chemical reactions, and embryo geometry. We assume that the fundamental diffusion and chemical reaction equations for both systems are similar, as described in (1), (2), (3). However, since the shape of the BMP signal differs greatly between Drosophila and zebrafish, we code these differences with variations in the biophysical parameters, as well as distinct geometries, length scales, and timescales of the two systems. These known differences are summarized in Table 2. The embryo length and the expression ranges of BMP, inhibitor (Chd/Sog), and Tld were estimated based on whole-mount experimental results for bmp2b, chd, and tld from the study by Li et al [70]. and Umulis et al [71]. The parameter space for the remaining unknown variables was defined using the ranges reported in the study by Zinski et al [74].

Table 2.

Parameters that specify the length and time scales of Drosophila and zebrafish.

Parameter	Drosophila	zebrafish	Best-Fitted Parameter set
Half embryo length	275μm	700μm
BMP expression	150μm (dorsal)	350μm (ventral)
Inhibitor expression	1μm (ventral)	150μm (dorsal)
Tld expression	150μm (dorsal)	450μm (ventral)
Time to establish gradient	≈ 1 h	≈ 2 − 3 h
BMP diffusion rate (DB)	10−2 ∼ 102 (μm2/s)		7.2671 (μm2/s)
Chd/Sog diffusion rate (DI)	10−2 ∼ 102 (μm2/s)		81.3047 (μm2/s)
BMP- Chordin/Sog complex diffusion rate (DBI)	10−2 ∼ 102 (μm2/s)		0.0260 (μm2/s)
Forward reaction rates for BMP and Chd/Sog (k1)	10−4 ∼ 100 (1/nM s)		0.0011 (1/nM s）
Reverse reaction rates for BMP-Chd/Sog (k_1)	= k1		0.0011 (1/nM s）
Decay rate of BMP (${\delta }_B$)	10−5 ∼ 10−1 (1/s)		0.0446 (1/s)
Decay rate of Chd/Sog (${\delta }_I$)	10−5 ∼ 10−1 (1/s)		0.0002 (1/s)
Decay rate of BMP-Chd/Sog complex (${\delta }_{\mathit{BI}}$)	10−5 ∼ 10−1 (1/s)		0.0002 (1/s)
Production rate of BMP (${\varnothing }_B$)	10−2 ∼ 102 (nM/s)		11.5602 (nM/s)
Production rate of Chd/Sog (${\varnothing }_I$)	10−2 ∼ 102 (nM/s)		96.9608 (nM/s)
Tld processing rate of Chd/Sog (${\lambda }_I$)	10−4 ∼ 100 (1/s)		0.0015 (1/s)
Tld processing rate of BMP-Chd/Sog complex $({\lambda }_{\mathit{BI}}$)	10−4 ∼ 100 (1/s)		0.0029 (1/s)

To evaluate the system across a broad parameter space, we conducted an initial search through the computational model-based screen of over 1,000,000 combinations of biophysical parameters of the major extracellular BMP modulators. We applied the Latin Hypercube Sampling (LHS) strategy wherein 11 independent parameters were selected from a uniform distribution in log space that spanned four orders of magnitude within the physiological range for each parameter. The parameter ranges used for sampling are listed in Table 2. For each sampled parameter set, we first performed an initial wild-type simulation; the model was then re-simulated with Chordin production set to zero to simulate the BMP signaling gradient in a sog(-/-) Drosophila mutant, and Chordin loss-of-function (LOF) scenario in zebrafish cases. All computations were performed using Purdue’s Supercomputing Clusters.

2.2. Parameter sensitivities

A Latin Hypercube Sampling Partial Rank Correlation Coefficient (LHS/PRCC) sensitivity analysis is employed to analyze how the objective function varies with model parameters to identify the most relevant parameters that influence data fitting. LHS/PRCC is particularly useful as it explores the entire parameter space of a model with a minimum number of model simulations [75]. PRCC is based on linear regression analysis and evaluates correlation coefficients between model inputs and outputs. The correlation coefficient is normalized between −1 and 1, which allows comparisons between model inputs. Herein, we focused specifically on the sampled points that are close to the Pareto front (Pareto local) to reduce the bias to the sensitivity caused by significant changes in the objective due to ill-fitting parameters. The sensitivity of the objective is analyzed on the points that are within the 95 % confidence interval of the Pareto front when the uncertainty is propagated from experimental data, as described in detail in Box 1.

Box 1. Pareto Front uncertainty.

Uncertainty in measurements propagates through objectives optimization, affecting Pareto fronts. To quantify this uncertainty in Pareto fronts, we developed a method based on Normalized Root Mean Square Error (NRMSE) as the objective function. The experimental data were defined as $\mathbf{d}=\overline{\mathbf{d}}+\epsilon$, where d ^¯ is a vector of mean data points and $\epsilon$ is a vector of normally distributed random variable, $\epsilon \sim \mathcal{N}\left(\mu ,\mathbf{V}\right).$ The objective function is defined as:

$$\mathit{RMSE}=\frac{\sqrt{{(y\left(\theta \right)-\overline{\mathbf{d}}+\epsilon )}^T(y\left(\theta \right)-\overline{\mathbf{d}}+\epsilon )}}{\sqrt{n}}$$

where $y\left(\theta \right)$is the model simulated by parameter vector, $\theta$ and $\mathit{n}$ is the number of data points in a dataset. Using a Taylor series, we linearized the objective function around the mean of the data, $\overline{\mathbf{d}}$, where $\epsilon =0:$

$$\mathit{RMSE}\left(\overline{\mathbf{d}}+\epsilon \right)\approx \mathit{RMSE}\left(\overline{\mathbf{d}}\right)+\frac{\partial }{\partial \epsilon }\mathit{RMSE}\left(\overline{\mathbf{d}}\right).\epsilon +\mathcal{O}\left(∥\epsilon ∥\right)$$

$$\frac{\partial }{\partial \epsilon }\mathit{RMSE}=\frac{{(y\left(\theta \right)-\overline{\mathbf{d}}+\epsilon )}^T}{\sqrt{n{(y\left(\theta \right)-\overline{\mathbf{d}}+\epsilon )}^T(y\left(\theta \right)-\overline{\mathbf{d}}+\epsilon )}}$$

Therefore, the linearized objective around the mean data is,

$$\mathit{RMSE}\left(\overline{\mathbf{d}}+\epsilon \right)\approx \mathit{RMSE}\left(\overline{\mathbf{d}}\right)+\frac{{(y\left(\theta \right)-\overline{\mathbf{d}})}^T}{\sqrt{n{(y\left(\theta \right)-\overline{\mathbf{d}})}^T(y\left(\theta \right)-\overline{\mathbf{d}})}}.\epsilon$$

which could be simplified to,

$$\mathit{RMSE}\left(\overline{\mathbf{d}}+\epsilon \right)\approx A+B.\epsilon$$

where $A,B$ are a constant scalar and vector, respectively that are independent on $\epsilon$. Since $\epsilon$ is normally distributed with mean, $\mu =0$ and covariance matrix, $\mathbf{V}$, we approximated the distribution of $\mathit{RMSE}(\overline{\mathbf{d}}+\epsilon )\sim \mathcal{N}(A+B\mu ,B\mathbf{V}{B}^T)$.

We introduced a novel concept of the Utopian frontier to compare the compromise front between the Drosophila and zebrafish systems, which is described in Box 2.

Fig. 2.

Illustration of the Utopian Front concept in multi-objective optimization. The blue curve represents the original Pareto front, connecting Anchor Point 1 and Anchor Point 2, which depicts the trade-offs between Objective 1 and Objective 2. The Global Utopian point (orange star) is an ideal but infeasible solution that simultaneously minimizes all objectives. Utopian Fronts are generated by re-optimizing each Pareto solution with a subset of variables while holding the complementary subset fixed.Subset 1 (green), Subset 2 (yellow), and Subset 3 (gray) each yield their own subset-specific Utopian points (Utopian point 1, Utopian point 2, and Utopian point 3). Connecting these Utopian points forms relaxed Utopian Front (red curve) that lie between the Pareto front and the Global Utopian point. These fronts reveal the improvement potential and objective sensitivity to specific parameter subsets in the optimization landscape.

Box 2. Utopian Front.

We introduce a novel concept of Utopian frontier which extends the concept of Pareto frontiers in multi-objective optimization. The Utopian frontier is defined a set of utopia solutions created by improving a subset of the unknown parameters while holding the rest at their Pareto optimal choices. Therefore, the Utopian frontier is derived from an existing Pareto frontier. The set of Pareto points is attained by solving a multi-objective optimization problem expressed as:

$$\underset{x\in X}{\min }f(x)=(f_1(x),f_2(x),\mathrm{·\; ·\; ·},f_n(x)),$$

where $f\in R^n$ are multi-objective functions of x constrained within a design space $\mathit{X;\; G}(x)\le 0$ and $X\in R^d$. A Pareto front consists of points, $x^{*}\in X$ that satisfy:

• 1.$f_i(x^{*})\le f_i(x)$ for all indices $\mathit{i}\in 1,2,...,\mathit{n}$ (strict dominance) and
• 2.$fj{(x}^{*})<f_j(x)$ for at least one index $\mathit{j}\in 1,2,...,\mathit{n}$ (weak dominance).

The set of all Pareto optimal design variables is denoted as$x*\in X^{*}$. A global utopian point is an idealized, theoretical infeasible point that simultaneously optimize all objectives functions.

As demonstrated in Fig. 2, for each$x^{*}\in X^{*}$, we re-optimized the objectives by varying only a subset, $x_s\in x^{*},$ while holding the remaining complement subset $x_s\prime \in x^{*},$ constant. This reduces the optimization search space to, $S=\left\{x_s\right\}\in R^{q<d}$ while the complement subset, $S\prime =\{x_{s\prime }\}\in R^{d-q}$ is treated as equality constraints. The new multi-objective optimization problem is then defined over the subset, $S$ for each Pareto solution, $x^{*}=(x_s,x_{S^{\prime }})$ as follows:

${x_s}^{*}=\underset{x_S\in S}{\mathrm{argmin}}{f}_j(x_s),j=1,...,n,$

subject to $x_{S\prime }=x_{S\prime }^{*}$

where $\mathit{j}$ are the n objective functions and $x_S^{*}$ is the set of solutions creating a new Pareto frontier that is tangent to the original Pareto front at the point, $x$. The Utopian point of the new Pareto frontier is the ideal point which could simultaneously optimize all objective functions, defined by $x_u$ as follows:

$$x_u\to (f_1(x_{S1}^{*},x_{S\prime }^{*}),f_2(x_{S2}^{*},x_{S\prime }^{*}),...,f_n(x_{\mathit{Sn}}^{*},x_{S\prime }^{*})),$$

where $x_{S_{\mathit{j}}}^{*}$are the anchor points for each $j^{\mathit{t}\mathit{h}}$ objective. The collection of all non-dominated Utopia points, $x_u\in X_u$forms the Utopian front solved along the original Pareto frontier, $x\in X$.

The Utopian fronts lie between the original Pareto front and the global utopia point. The convexity and distance of the of a Utopian front from the Pareto front reveal the relationship of subset variables and the trade-offs of the objectives in the Pareto front. A Utopia frontier that is invariant to the Pareto front suggests that differing optimal values of the subset variables for each objective cannot improve any of the objectives while a Utopian front close to the global utopia point reveals that differing optimal values of the subset variables for each objective could improve all objective functions.

3. Results

3.1. Fixed parameter common core model fails to simultaneously fit Drosophila and zebrafish systems under the same biophysical conditions

To evaluate whether the simplest core model, described in (1), (2), (3), could adequately describe both Drosophila and zebrafish BMP regulatory systems, we generated Pareto solutions by fitting the model to both systems with WT and mutant (sog(-/-) in Drosophila and Chd LOF in zebrafish) data simultaneously. The Pareto solutions obtained from fitting the WT data along and sog(-/-)/Chd LOF only and fitting the WT with mutant are shown in Fig. 3A, sampled from the parameter space in Figs. 3B and 3C. The Pareto front generated from WT data only exhibits high convexity, suggesting that there is a common set of biophysical parameters that fit WT Drosophila and zebrafish experimental data. The best compromise solution demonstrates high fitness to the WT data (blue line in Fig. 3A) with objective function penalties closer to zero (0.0242 and 0.0524 for Drosophila and zebrafish systems, respectively). However, this result seems to suggest a possible simple conserved mechanism between the two systems. Similarly, when fitting only mutant data (sog(-/-)/Chd LOF), the best-compromise solution for model simulations aligned well with experimental observations (yellow line in Fig. 3A), yielding residuals of 0.0367 and 0.0498 for Drosophila and zebrafish systems, respectively.

Fig. 3.

Core model Pareto Front analysis. (A) Pareto front generated by fitting the core model to WT, mutant (sog(-/-)/Chd LOF), and both WT and mutant data. (B) Core model simulation results space for WT fitting (C) Core model simulation results space for sog(-/-)/Chd LOF fitting. Corresponding model simulations with Drosophila anchor point (top panel), best compromise point (middle panel), and zebrafish anchor point (lower panel) are compared to (D)WT experimental data, (E) sog(-/-)/Chd LOF data The Chd LOF curve is not plotted in the figure due to an off-range result., And (F) both WT and mutant data. ExpDro and ExpZeb indicate experimental data for Drosophila and zebrafish, respectively, while SimDro and SimZeb indicate simulations for Drosophila and zebrafish, respectively. Mutant data indicate the sog(-/-) mutant in Drosophila data and Chd LOF in the zebrafish case for simplifying.

However, while the model performed well when fitting wild-type (WT) data (Fig. 3D) or mutant data (Fig. 3E) separately, it failed to reproduce both WT and mutant datasets simultaneously (Fig. 3F). When fitting both datasets together, the resulting Pareto front displayed reduced convexity compared to the WT-only or mutant-only fronts, suggesting a potential divergence in BMP signaling mechanisms between Drosophila and zebrafish that the core model does not capture. Although the model’s ability to fit mutant data improved, the best compromise solution still did not fully capture the BMP gradient features in Drosophila.

It is important to note that the fits were quantitatively evaluated using the root-mean-square error (RMSE) between the simulation outputs and the mean experimental profiles, rather than assuming a predefined error tolerance. While some model outputs, such as the example shown in the middle panel of Fig. 3F, appear visually along with the experimental data range, our multi-objective optimization framework focuses on exploring trade-offs across species rather than fine-tuning for a single best match. This approach enables a systematic analysis of parameter regimes that jointly satisfy both datasets, helping to reveal potential limitations in model generalizability. The inability to simultaneously reproduce WT and mutant data highlights the need for additional regulatory mechanisms or relaxed parameter constraints to more accurately model both systems under a unified biophysical framework.

Furthermore, to enhance the interpretability of the multi-objective optimization results, we extracted representative “best” parameter sets from the Pareto front and performed a localized perturbation analysis. Specifically, each parameter was varied individually within its feasible range while holding the others fixed at their best-fit values. This one-at-a-time approach allowed us to visualize the impact of individual parameters on model performance and assess the local curvature of the Pareto front in parameter space. The resulting parameter sensitivities and solution shifts are presented alongside the main optimization results in Fig. 3, providing clearer insight into how specific biophysical parameters contribute to model trade-offs. A complete list of the best-fit parameter combinations is listed in Table 2, column 3. Notably, in our results, the best-fit parameter set that simultaneously reproduces BMP gradient profiles in both zebrafish and Drosophila embryos requires a high Chordin production rate (∼96 nM/s). While this value exceeds experimentally inferred rates, we interpret it as a compensatory mechanism within the simplified model framework. In vivo, such strong BMP antagonism may not arise solely from high Chordin expression, but rather through additional mechanisms such as spatially restricted production, shuttling interactions (e.g., with Tsg or Sog), or stabilized Chordin–BMP complexes. This result highlights that while our core model can capture conserved gradient features across species, differences in regulatory architecture may allow each system to achieve similar outcomes through distinct mechanistic routes.

3.2. Chordinlike variants shift the Drosophila system closer to zebrafish while preserving shuttling

In a key rescue experiment, ‘Chordinlike’ variants introduced into sog(−/−) Drosophila embryos significantly broadened the steep BMP gradient observed with endogenous Sog [21]. Notably, Peluso et al [21]. found that while Sog requires BMP for efficient cleavage by Tld, Chd can be cleaved by Tld in the absence of BMP. This fundamental difference in cleavage specificity helps explain how the Drosophila gradient can be shifted to resemble that of vertebrate embryos while maintaining a degree of BMP shuttling [76], [77], [78]. To implement this phenomenon, we specified the Tld cleavage rate of ’Chordinlike’ Sog to be slightly higher than that of endogenous Sog and investigated whether the broader BMP gradient observed with ’Chordinlike’ Sog could indicate a shared regulatory mechanism between Drosophila and zebrafish. The resulting Pareto front generated by fitting the model to "Chordin-like" data dominates the Pareto front of the endogenous system (Fig. 4A). The differences in the Pareto fronts are particularly pronounced along the trade-off regions, suggesting that with the "Chordin-like" Sog, the Drosophila system shifts towards a BMP mechanism similar to zebrafish. To ensure that the observed differences in Pareto fronts were not simply due to propagated uncertainty, we applied an uncertainty propagation technique. Figs. 4A, 4C, and 4E illustrate how the uncertainty propagates from experimental measurements to the objective space for both models. The error bars represent 95 % confidence intervals, and their non-overlapping nature confirms that the Pareto frontier differences are not simply due to experimental uncertainty. Instead, they reflect fundamental changes in the regulatory dynamics, proving that the observed shift in BMP regulation is due to the biophysical difference between endogenous Sog and "Chordin-like" Sog variants.

Fig. 4.

Comparison of endogenous and ‘Chordin-like’ Core Model Pareto fronts. The ’Chordin-like’ Pareto front dominates the endogenous front, demonstrating that ’Chordin-like’ data shifts the Drosophila BMP mechanism to resemble that of zebrafish for both the WT fitting (A), ’Chordin-like’ sog (-/-)/ Chd LOF mutant fitting (C) and WT+ ’Chordin-like’ sog (-/-)/ Chd LOF mutant fitting (E). Corresponding model simulations of the best compromise point for both Drosophila and zebrafish are compared to ‘Chordin-like’ experimental data. (B) Compromise solution for ‘Chordin-like’ Drosophila data vs WT zebrafish data. The CLF curve is not plotted in the figure due to an off-range result. (D) Compromise solution for ‘Chordin-like’ Sog(-/-) mutant Drosophila data vs Chd LOF zebrafish data. (F) Compromise solution for ‘Chordin-like’ + ‘Chordin-like’ Sog(-/-) mutant Drosophila data vs WT+Chd LOF zebrafish data.

Surprisingly, despite the biophysical differences between Sog and Chd revealed by the “Chordin-like” experiment, BMP shuttling in Drosophila remained intact. As shown in Fig. 4D, BMP movement toward the dorsal midline was still observed, indicated by a reduced BMP gradient in sog(-/-) embryos compared to WT. Although shuttling was preserved, its strength was diminished. The WT BMP gradient remained broader, and the sog(-/-) simulation shows only about a 20 % decrease. Furthermore, the slight increase in the Tld processing rate of free Sog corroborates previous findings that Sog processing requires BMP, whereas Chd can be cleaved by Tld independently of BMP [21]. Finally, the Pareto front generated by the “Chordin-like” mutant (Fig. 4C) aligns more closely with Drosophila sog(-/-) data than with zebrafish Chd LOF data, underscoring the unique biochemical properties that differentiate Sog from Chd.

3.3. Differential parameter ranges reveal species-specific trade-offs in BMP signaling

To determine the influence of individual parameters on the Pareto front, we evaluated how the 11 independent biophysical parameters affect the solution space and Pareto fronts. We examined 1,000,000 simulation results in RMSE space for both Drosophila and zebrafish under both WT (Fig. 5A) and mutant conditions (Fig. 5B), with a color scale reflecting the specific range for the individual parameter. We observe that specific parameters, particularly those linked to Chordin/Sog activity, play a prominent role in shaping the Pareto front. In the WT scenario (Fig. 5A), the diffusion rates of BMP (D_B), Chd/Sog (D_I), and the BMP–Chd/Sog complex (D_BI), as well as the production rates of BMP (${\mathrm{\varnothing }}_B$) and Chd/Sog (${\mathrm{\varnothing }}_I$), tend to cluster in regions closer to the Pareto point. A similar trend is observed under sog(−/−)/Chd LOF conditions, but with an additional emphasis on λ_BI (the Tld processing rate of the BMP–Chd/Sog complex), which becomes more pronounced in mutant simulations. We further evaluated how changes in these parameter ranges shift the Pareto front (Fig. 6). Notably, increasing the diffusion rate of Chd/Sog (DS) consistently draws the Pareto front towards an optimal Pareto point (Fig. 6A). This effect is more pronounced when DS falls within the range of 10 −2 to 10–1 (μm2/s). Additional simulations using the “Chordin-like” variant (Supplementary Figures S1) show analogous parameter trends and confirm that elevated diffusion rates and processing factors shift the Pareto front toward an optimal region, reinforcing our primary findings. These findings suggest that while specific parameters share broad optimal ranges across Drosophila and zebrafish, others—especially those related to diffusion—may exhibit species-specific constraints, indicating evolutionary divergence in BMP regulatory mechanisms between the two species.

Fig. 5.

Parameter trend analysis for all the simulation results against WT and Chd/Sog(-/-) fitting. Figure group A shows the parameter tendency in Pareto space against WT fitting for 11 independent parameters, including A1, k₁ (forward reaction rates for BMP and Chd/Sog); A2, ${\delta }_B$ (Decay rate for BMP); A3, ${\delta }_I$ (Decay rate for Chd/Sog); A4, ${\delta }_{\mathit{BI}}$ (Decay rate for BMP-Chd/Sog complex); A5, D_B (Diffusion rate for BMP); A6, D_I (Diffusion rate for Chd/Sog); A7, D_BI (Diffusion rate for BMP-Chd/Sog complex); A8, ${\lambda }_I$ (Tld processing rate of Chd/Sog); A9, ${\lambda }_{\mathit{BI}}$ (Tld processing rate of BMP-Chd/Sog complex); A10, ${\varnothing }_B$ (Production rate of BMP); A11, ${\varnothing }_I$ (Production rate of Chd/Sog). Group B shows the parameter tendency in Pareto space against Chd/Sog(-/-) fitting for 11 independent parameters, respectively. Notably, the RMSE range is limited to [0,0.6] for Zebrafish fitting and [0,0.8] for Drosophila fitting for better visualization. The color represents the range of the individual parameter accordingly.

Fig. 6.

Comparison of best-fitting parameter range distributions for WT+mutant BMP gradient profiles in zebrafish and Drosophila. Boxplots show the distributions of parameter values among the top 1 % of parameter sets (lowest RMSE) that best fit the wild-type data for each species. Parameters are plotted on a logarithmic scale. Blue and orange boxes represent zebrafish and Drosophila, respectively. Diffusion-related parameters (D_B, D_I, and D_BI), processing parameter λ_BI and inhibitor expression Φ_I exhibit clear divergence between the two systems. In contrast, Drosophila solutions cluster at lower diffusivity values of BMP. Other parameters, such as decay rates and signaling thresholds, show more overlap between species, suggesting a greater degree of conservation.

To further examine how parameter regimes differ across species when fitting both wild-type and mutant phenotypes, we analyzed the top 1 % parameter sets that yielded the lowest combined RMSE for Drosophila and zebrafish. Fig. 6 presents the distribution of these best-fitting parameters for each species on a log scale. Several trends emerge from this comparison. Most notably, diffusion-related parameters, D_B, D_I, and D_BI, exhibit divergence: zebrafish favors significantly higher diffusion rates for all three species, consistent with its broader BMP gradient and larger embryonic geometry. In contrast, Drosophila solutions cluster around lower diffusivity values, aligning with the steep and spatially confined BMP profiles observed in vivo. In addition to diffusion, Zebrafish tend to favor lower values of processing parameters λ_BI and inhibitor expression ϕ_I, indicating that proteolytic regulation may be differentially tuned across species to accommodate distinct gradient shapes. ϕ_I differs substantially, which may suggest that the two systems may rely on distinct sensitivities to inhibitory signals. Other parameters, including degradation rates (δ) and output thresholds (ϕ_B), show more overlap between species, suggesting these aspects of the network may be under stronger evolutionary constraint or serve more conserved regulatory roles. Taken together, this analysis reveals that the most robust cross-species fits emerge from differential tuning of biophysical parameters, particularly those related to extracellular transport and processing. These findings support the broader conclusion that while the BMP network architecture is conserved, its quantitative regulation is system-specific—tuned to match the geometric and developmental context of each species.

3.4. Relaxed model results show Utopian fronts that favor processes that impact morphogen mobility between Drosophila and zebrafish

The BMP gradient relies on the balance of geometry, biophysical interactions, and conditions of the systems to establish observed BMP gradients in both Drosophila and zebrafish. Herein, we take a step-by-step investigation to determine the minimum combination of biophysical interactions and parameter conditions needed to generate the differences in the mechanisms. A Utopian front outperforms the original Pareto front, suggesting that allowing the biophysical parameters to differ between Drosophila and zebrafish improved the model’s ability to fit both sets of data. To identify the influential parameter combination that improves fit for both systems, we tested our model by introducing a relaxation scheme based on our prior sampling of model parameters. These relaxation tests were conducted using the parameter sets from the original Pareto front obtained in the initial screen of 1,000,000 parameter sets, where both systems shared identical parameters (considered a relaxation degree of zero). This approach introduces additional degrees of freedom (DoF) in the systems by allowing subsets of shared parameters to vary between Drosophila and zebrafish. Specifically, in the one-degree scenario, a single parameter was allowed to vary between the Drosophila and zebrafish models, while all others remained constant and identical in both systems. This process was repeated for all 11 parameters. The two-degree scenario involved allowing two parameters to vary between the systems, producing 55 unique combinations, while the three-degree scenario produced 156 possible combinations of relaxed parameters.For each scenario and unique parameter combination, an additional subset optimization was performed with 100 samples each for zebrafish and Drosophila, centered around original Pareto points in the WT, mutant, and WT+mutant fits. Our results revealed a more prominent Utopian front as the DoF increased in WT only (Fig. 7A), mutant only (Fig. 7B), and WT + mutant conditions (Fig. 7C). For WT and mutant fronts, relaxing two parameters was sufficient to closely approach the Utopian front, whereas for WT + mutant fitting, relaxing all three diffusion-related parameters (D_B+ D_I +D_BI) yields the most significant improvement.

Fig. 7.

Utopian Fronts derived from subset-variable relaxation across three optimization contexts. (A) Wild-type (WT) Pareto front. (B) Sog(-/-)/Chd loss-of-function (LOF) mutant Pareto front. (C) Combined WT + Sog(-/-)/Chd LOF fit front. Each Utopian front is constructed by relaxing a subset of parameters from the globally optimized Pareto set (degree 0) and re-optimizing only those parameters while holding the others fixed. Colored lines indicate the subsets that most advance the front toward the utopia point. Blue solid line show the most influential single-parameter (Degree 1) relaxation (δ_B in WT fit, D_B in mutant fit and K₁ in WT+mutant fit), Green solid line show the most influential two-parameter (Degree 2) combinations (δ_B + λ_BI for WT fit, δ_B + δ_BI for mutant fit, D_I + D_BI for WT+mutant fit), Red dash line show the most influential three-parameter (Degree 3) combinations (δ_B + λ_BI + D_BI for WT fit, δ_BI + D_B + ϕ_I for mutant fit, D_B+ D_I +D_BI for WT+mutant fit). The black solid line shows the original Pareto front from the initial screen. For WT and mutant fronts, relaxing two parameters is sufficient to closely approach the Utopian front, whereas for WT+mutant fitting, relaxing all three diffusion-related parameters yields the most significant improvement.

To further investigate which parameter combinations most strongly contribute to the improved fit observed in the Utopian front, we systematically analyzed the outcomes from parameter relaxation scenarios involving one, two, and three degrees of freedom (DoF). In each case, we compared the resulting Utopian fronts under different parameter combinations to determine which configurations most effectively enhanced model performance across systems. The complete set of results is shown in Fig. 8, where panels A1–A3 correspond to one-degree-of-freedom relaxations, B1–B3 to two degrees, and C1–C3 to three degrees. In each panel, the highlighted Utopian front indicates the best-compromise solution achieved under that relaxation level, while the black front represents the original Pareto front without any parameter relaxation.

Fig. 8.

Systematic analysis of Utopian front improvements under increasing degrees of parameter relaxation. (A1–A3) Utopian fronts generated by relaxing one parameter at a time (1 degree of freedom, DoF) between Drosophila and zebrafish models. The most impactful parameters (highlighted in blue) are BMP decay rate (δ_B) for WT, BMP diffusion rate (D_B) for mutant, and BMP–inhibitor binding constant (K₁) for WT+mutant fits. (B1–B3) Utopian fronts with two parameters relaxed (2 DoF). Optimal combinations (green) include δ_B with BMP–inhibitor complex processing rate (λ_BI) for WT, inhibitor decay (δ_BI) with inhibitor diffusion (D_I) for mutant, and inhibitor diffusion (D_I) with BMP–inhibitor complex diffusion (D_BI) for WT+mutant fits. (C1–C3) Utopian fronts with three parameters relaxed (3 DoF). Key combinations (red) are δ_B, λ_BI, and D_BI for WT; δ_B, δ_BI, and inhibitor production rate (ϕ_I) for mutant; and D_B, D_I, and D_BI for WT+mutant fits. In all panels, highlighted colored lines indicate the best-compromise parameter combinations, black curves represent the original Pareto fronts without relaxation, and black shaded lines indicate other parameter combinations that did not yield a substantial shift toward the Utopian front. Results show that for WT and mutant fits, relaxing two parameters is sufficient to approach the Utopian front, whereas for WT+mutant fits, relaxing all three diffusion-related parameters yields the greatest improvement. Diffusion-associated parameters consistently shape the Utopian front, suggesting that variation in effective diffusivity is a major factor in species-specific BMP patterning strategies.

For both WT and mutant datasets considered separately (Fig. 7A, 7B), relaxing just two parameters was sufficient to advance the Pareto front substantially toward the utopian point, indicating that relatively small shifts in a few key biophysical properties can explain much of the interspecies difference. In WT, these parameters correspond to the BMP decay rate (δ_B) and the cleavage rate of the BMP–inhibitor complex (λ_BI), suggesting that tuning ligand stability and inhibitor processing can reproduce species-specific BMP profiles. In the mutant case, the most effective changes involved the BMP diffusion rate (D_B), pointing to the importance of BMP mobility and clearance when Chordin/Sog function is absent. In contrast, the mutant front showed minimal improvement across all three degrees of relaxation; slightly tuning for the utopia front is not very meaningful in the mutant scenario, indicating that the simple core model can adequately fit WT conditions but fails to reproduce the mutant phenotype. This suggests that additional species-specific regulatory extensions—such as alternative feedback or compensatory pathways—are likely required to capture mutant-specific BMP signaling dynamics. In contrast, the combined WT+mutant fit (Fig. 7C) required relaxing three diffusion-related parameters (D_B, D_I, D_BI) to achieve the most significant improvement, highlighting that coordinated adjustments in the mobility of BMP, its inhibitor, and their complex are essential to capture both phenotypes simultaneously. This trend suggests that system-specific tuning of diffusivity is a key mechanism by which regulatory networks may adapt to the distinct spatial and developmental constraints of individual organisms. Notably, the identification of diffusion-related parameters (D_B, D_I, and D_BI) as the primary drivers of Utopian front advancement is consistent with our earlier analysis of best-fitting parameter range distributions (Fig. 6), which showed clear divergence in these parameters between Drosophila and zebrafish. While our model represents these effects through classical diffusion terms, in biological systems, such behavior may emerge from more complex processes such as the shuttling of Sog/Chordin by cofactors like CV2, which facilitates long-range transport and effective expansion of the morphogen's spatial influence. These findings aligned with the goal of our analysis, which is not to make direct predictive claims, but rather to establish a framework that enhances the interpretability and cross-system applicability of mechanistic models. The improved fit of the Utopian front demonstrates the value of relaxing assumptions of strict parameter conservation when comparing divergent biological systems, offering a more translationally relevant modeling strategy across species.

4. Conclusion

Our work highlights the use of multi-objective optimization (MOO) and introduces a novel concept of the Utopian front, an extension of Pareto fronts, to analyze the conservation and diversification of the BMP mechanism between Drosophila and zebrafish. By applying Pareto front analysis, we explore how the biophysical properties of the BMP pathway have been conserved in Drosophila and zebrafish systems. We identified minimal biophysical requirements that reconcile known geometric and temporal differences between Drosophila and zebrafish while producing experimental observations. Through the Pareto analysis, we identified a set of biophysical parameters that generated the best compromise to fit both Drosophila and zebrafish BMP systems.

Experimental data further suggest a divergence between the Drosophila and zebrafish BMP pathways. The novel concept of Utopian fronts provides a new framework to (i) assess system constraints in fitting different objectives and (ii) distinguish between variant and invariant components within a system. By combining Pareto optimization and Utopian analysis, we systematically varied a subset of parameters (variant component) while keeping others constant at their Pareto optimal values (invariant). Geometric differences alone proved insufficient to explain the evolutionary divergence of BMP mechanisms. Instead, a balance between geometry and biophysical properties was required to produce BMP patterns in Drosophila and zebrafish. Instead, our findings emphasize that a combination of geometry and species-specific biophysical tuning—particularly in diffusion and processing parameters—is required to recapitulate BMP patterns in both systems.

To enable a precise and quantitative comparison of extracellular BMP signaling across species, we focused on a core network consisting of BMP ligands and their primary extracellular inhibitors. The Utopian analysis revealed that diffusion-related parameters, including the diffusivity of BMP, its inhibitor, and the BMP–inhibitor complex, consistently shaped the best-compromise solutions. These findings suggest that differential diffusivity may serve as a flexible regulatory lever across systems. Furthermore, parameters associated with diffusion activity emerged as key drivers of BMP pattern divergence. While our model treats this behavior as classic diffusion, it may reflect underlying biological mechanisms such as Sog/CV2-mediated shuttling that achieve long-range morphogen transport. These parameters, previously shown to be important in BMP shuttling in Drosophila, help generate the steep and sharp BMP gradient that is evidently different from the zebrafish BMP mechanism. The evolutionary changes, especially in BMP-specific parameters, in addition to differences in geometry and complex incorporation of positive feedback (in Drosophila) and Noggin regulation (in zebrafish), contribute to species-specific BMP mechanisms. By applying a parameter relaxation strategy with increasing degrees of freedom, we demonstrated that even minimal changes to a small set of diffusivity-related parameters significantly improved cross-species model performance. Expanding the model to include additional species-specific components—such as CV2, Tld regulation, or intracellular feedback—is a promising direction for future investigation.

Overall, our findings demonstrate how Pareto optimization and Utopian analysis provide a structured approach to studying complex systems. This framework not only helps investigate evolutionary system dynamics but also identifies components of systems selected for conservation and those chosen for divergence. More broadly, this methodology offers a powerful tool for dissecting systems-level behavior in other biological contexts where mechanisms can be compared through a shared model structure or parameter space.

CRediT authorship contribution statement

Linlin Li: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. David Umulis: Supervision, Methodology, Conceptualization. Thembi Mdluli: Writing – review & editing, Writing – original draft, Software, Methodology, Data curation, Conceptualization. Gregery Buzzard: Supervision, Methodology, Conceptualization.

Declaration of Competing Interest

None.

Appendix A. Supplementary material

Supplementary material

Data Availability

All MATLAB code used for simulations, data processing, and figure generation in this study is available at: https://github.com/linlinli12/MOO.