Dynamic systems theory in biology

Network science has experienced an enhanced interest over the last years with a lot of research involved in understanding the complexity and causality within network interactions that drive many natural and artificial systems [1,2]. In biology, the measurement and collection of large-scale molecular data such as genomics, transcriptomics, proteomics and metabolomics [3,4] has imposed a growing need for new methods and tools to integrate and analyse multiomics interactions in ways that follow the real underlying mechanisms within molecular networks [5–7]. Often, purely statistically based machine learning tools are limited in their capacity to identify true mechanistic associations in the data due to relying on variances and correlations that might be spurious and not causal in nature. In contrast, tools that base their assumptions on a predefined model allow for a hypothesis on causality relations to be made that can be refuted or approximated based on prediction or reconstruction metrics. Dynamical systems theory is a mathematical domain that offers a plethora of such tools that contain a lot of generic formalisms, transferable within biological domains [8].

Widely used and well established in the fields of control engineering and physics, dynamical systems have also experienced a surge within biology from quantifying the dynamics of gene expression [9], to finding differential biochemical reaction kinetics [10], to generating distinct dynamical behaviours via toggle switches [11] and switching cancer cell types [12]. The plethora of applications acknowledge the dynamic nature of biological systems such as ecological networks, embryonic pattern formation and interactions within molecular networks [8]. Thus, there is a shift from static representations to increasingly dynamic accounts of biology [13,14]. This shift needs to be matched with a transition from the use of static statistical tools to dynamical systems tools that consider highly non-linear interactions in biology. Current challenges with ODE models for studying dynamical systems in biology is that the tools are mostly tailored for low dimensional systems [15] but usually ecological and multiome molecular data are high-dimensional (small sample sizes, large feature spaces) [8]. They also require a priori knowledge of kinetic equations with many unknown parameters [16] which are hard to define in high-dimensional biological systems.

Dynamic mode decomposition as a data-driven dynamical system tool

Here we aim to explore the applicability of a data-driven dynamical systems tool, Dynamic Mode Decomposition (DMD) (see Methods), in the network integration of multiome short time-series data of two Clusia species, C. rosea and C. major that exhibit alternate forms of plant photosynthesis. DMD is a well-known time series data-reduction approach, that approximates forward dynamics discretely in time [17] and is applicable to big datasets such as omics data with many unknown parameters. It was originally developed for decomposing high-dimensional data into coherent temporal components, also known as eigenmodes, that represent distinct linear trajectories in time. At the same time, it approximates the Koopman operator that allows to perform diagnostics of the lower-dimensional linear approximation model of the data. DMD combines the spatial data reduction step of principal component analysis (PCA) and time decomposition step of the discrete-time Fourier transform (DFT) [18]. Entirely based on measurement data and with no need to define an explicit physical model, it has been applied in many different fields such as fluid mechanics [19], climate and geosciences [20], biochemical systems [21], robotics [22], power grids [23] and economics [24]. The eigenmodes that are a result of DMD, can be causally linked to different behaviours of a system and provide physical interpretability, regime detection and classification. For example, DMD applied to functional magnetic resonance imaging (fMRI) scans resulted in clusters of eigenmodes that corresponded to resting state networks (RSNs) of different brain regions [25]. Furthermore, the linear Koopman operator can be used as input to optimal control strategies that steer the direction of the system to a desired outcome [26–28].

Data-driven approaches have emerged as effective tools to bypass the need of assumption-based forward models and corresponding parameters [8,29,30]. We previously implemented a data-driven inverse modelling approach using the stochastic Lyapunov matrix equation to reveal control points in highly complex metabolomics data by Jacobian reconstruction [8,29,31]. This allowed the identification of a novel metabolic checkpoint for macrophage polarization [32] which eventually was proven to control tumor-associated macrophages and tumor growth [33,34]. Generic principles of the inverse stochastic Lyapunov equation can be applied universally in biology and ecology [8]. These methods are fundamental for understanding biological systems because they reconstruct realistic models directly from empirical data, minimizing the reliance on an a priori parameterized model [8]. By using DMD, we overcome certain limitations of the Jacobian reconstruction while extending to non-linear and higher order dynamics (see kernel-DMD in Methods) within the bounds of the sampled time series of a dataset. With the combinatorial approach we use here, more than one omics data type can be integrated into dynamic network representations without the need of a-priori knowledge of network interactions as required for the Jacobian reconstruction. The method also extends to bigger datasets as found within transcriptome or proteome data due to its data reduction step, for which the Jacobian derivation becomes more unstable with increasing feature size [35]. While the Jacobian operates on steady-state data using covariance matrices across different conditions, DMD is applied on time series and discovers time-dependent dynamics. Therefore, DMD for multiome network integration is a significant extension of the data-driven dynamic toolset.

A limitation of using dynamical systems tools, or any tool borrowed from different scientific disciplines, is whether they are appropriate in providing a holistic mechanistic understanding of molecular systems for which the governing equations are not explicitly known [36]. However, even though a chosen model might not represent the full reality of a biological system, it can at least provide knowledge of some of its properties such as stability, modularity, bifurcation parameters or emergence and is therefore suitable for specific applications such as input control. Within the kernelized form of DMD, the type of nonlinear interactions between variables can be approximated using a kernel. Here we used a kernel-DMD variant as introduced in LANDO [37] which overcomes the limitations of linear DMD for nonlinear systems such as multiome networks. The kernel enables implicit physical assumptions to be made in the interactions of features via the use of the dot product which can be embellished within simple tailored functions such as the polynomial or gaussian functions. Various kernel-DMD approaches have been developed [38–41] and evaluated in their application to various nonlinear systems that showed better performance and Koopman spectra than classical DMD [40,42–45]. We chose LANDO because it circumvents the need to explicitly define observables in the higher-dimensional space, such as SINDy [38], which is computationally expensive. Instead it scales the problem to the dimension of the time component of the series and not feature size, which is particularly fitting to molecular data with high feature size and low sample size. Contrary to other kernel-DMD efforts, LANDO maps the kernel and its learned weights to the original state values, which enables the extraction of the linear Koopman operator in the original feature space providing more physical interpretability of the feature interactions in the Koopman network. This is helpful for concrete conclusions to be made in terms of molecular networks. It also disambiguates the linear from non-linear components, which we showcase in this paper to be useful for different downstream applications such as the eigendecomposition emerging from the linear part and control mechanisms from the nonlinear part. Here we take advantage of the kernel method, to tailor a “Hill function”-like kernel, via a combination of sigmoid and linear kernels (see Methods section below). Hill function dynamics are frequently used to model molecular network interactions [46] and the hypothesis is that they are also appropriate for modelling multiome data interactions. The use of a coarse-grained kernel DMD data-driven model, is an efficient way to circumvent the problem of high-dimensionality in datasets where an interaction model between all sets of features is impossible to know a priori.

Application to differential fluxes between different types of photosynthesis

We demonstrate the suitability of multiome Koopman integration by identifying dynamical differences between two types of plant photosynthesis (Crassulacean acid metabolism (CAM) and C3-like, S1 Fig). CAM is a type of photosynthesis that has physiologically evolved in adaptation to low water availability. During CAM, stomatal opening occurs in the night instead of the day, resulting in nocturnal uptake of CO₂. Phosphoenolpyruvate carboxylases (PEPCs) carboxylate Phosphoenolpyruvate with CO₂ eventually stored as organic acids such as malate or citrate within the vacuole. During the day decarboxylation of stored organic acids restore CO₂ levels, thereby ameliorating the need for stomatal opening and minimizing water loss. C. rosea, a well-known constitutive CAM species [47], differs from its close relative, C. major, which is a C3-like plant.

Here, we aim to uncover the differences in the subtypes of photosynthesis within these two plant species using our combinatorial Koopman approach for network integration (see workflow Fig 1). Firstly, we evaluate different kernels (Fig 1A2) and their hyperparameters to evaluate whether the sigmoidal or “Hill function”-like kernel is a good fit to both forward dynamics and output measurements in a multi-objective function. We use the Taylor series expansion to extract the linear and nonlinear parts of the model. The linear part is effectively the Koopman operator (Fig 1A3) of which the eigendecomposition results in eigenmodes that correspond to different temporal dynamics. The distinction between weak versus strong CAM photosynthesis becomes clearer once we relate the predictive capacity of selected eigenmodes to stomatal opening CO₂ gas exchange measurements – a distinctive physiological signature between C3 vs CAM due to opposing stomatal opening trends (Fig 1A4). We also introduce new input control mechanisms in C. major for switching between photosynthetic types and phenocopying the C. rosea output signature (Fig 1B). This comprises of several steps starting from the identification of appropriate control matrix and inputs making use of the nonlinear part of the learned model (Fig 1B1), to reducing the dimensionality of the data (Fig 1B2), carrying out alignment (Fig 1B3) and then performing linear Model Predictive Control (MPC) in the reduced subspace (Fig 1B4). The linearization of the system first by Taylor series expansion to get the Koopman operator and then SVD of the nonlinear part of the Taylor series to obtain the linear control matrix B, was crucial for making the system amenable to linear MPC, which requires a linear model [48]. The data reduction step was also an important prerequisite to linear MPC because the latter failed to predict good output reconstruction accuracies in the full feature space. It was therefore imperative to find a method that selected the most important features in terms of dynamics in order for MPC to work efficiently with regard to time and computational complexity [49]. We also found that the alignment step through the use of a transformation matrix was also a necessary prerequisite to MPC for proper alignment of Koopman matrices between the two systems that at the same time could find the right projection between output spaces.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Illustration of kernel-DMD with control workflow applied to multiome timeseries data of two Clusia species.

A) System Identification using kernel-DMD. A1) Multiome timeseries dataset collection (transcriptome, proteome and metabolome) of two Clusia species C. major and C. rosea with different photosynthetic physiotypes (C3-like vs strong CAM respectively). Time snapshot rearrangement for input to kernel-DMD where X includes time intervals 1 to m-1 and X’ one time step ahead (time interval 2 to m) with m the total number of time points. A2) The kernel applied to the concatenated multiome X, implicitly embeds the original features in the Reproducing Kernel Hilbert Space (RKHS) which includes non-linear interactions. This way we avoid an exhaustive search for optimal parameters of inner products in the higher-dimensional nonlinear feature space . A3) Linear Koopman model extraction results in coherent dynamics described by the eigenvalues of the Koopman operator. The model is evaluated based on the linear data and output reconstruction accuracies (R² score). This enables a hyper-parameter search for optimal kernel choice and its parameter values in step A2. A4) After fitting kernel--DMD to both species, their eigendecompositions are compared and the eigenmodes that best predict CO₂ gas exchange data (as a physiological signature of opposing CAM trends) are stored. Their most frequent edges are retained into a consensus eigenmode topology for which a glycolysis module is more dominant in C. rosea and a photorespiration module in C. major. B) System Control for the phenocopy of CAM output signature as exhibited in C. rosea within a C3  molecular background as in C. major. B1). The Taylor series expansion of the kernel-DMD model disambiguates the linear from nonlinear parts. The nonlinear part N(x) can act as a forcing constraint to the linear part. Its reduced control basis via Singular Value Decomposition (SVD) leads to the B control matrix and control inputs u. B2) Hankel Singular Values are used for feature selection as a way to select the most observable and controllable features for data reduction. B3) System re-identification in the reduced feature space is followed by projection of the C. major Koopman operator to that of C. rosea while at the same time ensuring alignment of their output dynamics. B4) The aligned linear Koopman matrices are used as input to Model Predictive Control where phenocopy is achieved for predicting new outputs in C. major that closely resemble those in C. rosea by finding optimal control inputs u. B5) Sensitivity analysis identifies positive feedback applied to glycolytic enzymes and negative feedback applied to photorespiratory enzymes. All images in Fig 1 were drawn by hand, except the photos of C. major and C. rosea in Fig 1A which were taken by colleagues who gave full permission to publish them under the CC BY 4.0 license. Images of the transcriptome, proteome and metabolome in Fig 1A were produced by ChatGPT5.

https://doi.org/10.1371/journal.pcbi.1014029.g001

To our knowledge, our work is one of the first attempts to validate kernel-DMD with subsequent optimal control in a high-dimensional multiome dataset and its capability for phenocopying. Our methodology has impact in the fields of crop science that aims to increase water use efficiency of C3 crops under arid, harsh conditions [50] by introducing new emerging bioengineering possibilities of advanced mechanisms of photosynthesis such as CAM. With this approach we also ameliorate limitations of limited sampling points obtained during typical measurements in ecological and biological systems. DMD discretely maps each time snapshot to the next without the requirement of significant time lag within the time series. Below we outline the methodology and workflow that uses the DMD formulation as a backbone.

Data and data preprocessing

Our dataset is high-dimensional (23 timepoints and 390 features) and includes 3 omics types: transcriptome, proteome and metabolome. Throughout the paper we make comparisons between C. major (represents C3-like photosynthesis) and C. rosea (represents strong CAM photosynthesis) (Fig 1A1). Transcripts with a TPM value below 1 were filtered out as well as low variant features (variance threshold < 0.3) in the proteome and transcriptome. All metabolites and CAM-related features according to our previous pathway annotation, were included in both proteome and transcriptome as well as the features corresponding to highest loading scores of the first six Principal Components after PCA data reduction. Any features not found in a species were 0-padded. Protein and transcript data retained were in equal proportions while the metabolome was less represented in terms of metabolites measured. After addition of a small random noise in each feature column to extend the timeseries from 12 to 24 samples, the concatenated multiome was z-transformed around mean 0 and standard deviation 1. The samples were rearranged such that the timeseries was in consecutive timesteps (4am, 8am, 1pm, 7pm) with replicates serving as distinct points in the series.

Dynamic mode decomposition and the Koopman operator

We consider dynamical systems that can be expressed as ordinary differential equations as in

(1)

where describes the continuous-time evolutionary dynamics of an -dimensional vector that characterizes the state of the system at time t, more specifically the concentrations of concatenated multiome features after z-transformation. F gives information on further tasks such as physical understanding, prediction and control. Due to the high non-linearity of dynamic systems, one major goal is to identify a new vector of coordinates , such that the time-evolution can be represented in linear terms. In practice, data is sampled discretely in time, so the aim is to find a matrix A, known as the Koopman operator [51,52], that advances dynamics discretely in time as in

(2)

The eigendecomposition of the Koopman operator completely characterizes the dynamics of the linearised system.

In its simplest form, Dynamic Mode Decomposition (DMD) is a data-driven method that approximates the Koopman operator from discretely sampled data. The snapshot data is arranged into two matrices and :

(3)

where m is the number of time steps. The Koopman DMD approximation in terms of these matrices is

(4)

The best fit A that advances snapshots linearly forward in time can be formulated as an optimization problem

(5)

where denotes the pseudo-inverse. The pseudo-inverse can be computed using Singular Value Decomposition (SVD) as . In practice, the effective rank of the data matrices after SVD is lower than their full dimension, so A can be projected to a lower dimension using the first r Principal Orthogonal Decomposition (POD) modes of U, approximating the pseudo-inverse using the rank-r SVD approximation :

(6)

Spectral properties of the Koopman operator give a complete diagnosis of the dynamical system with the description of the dynamic flow by the linear combination of dominant coherent structures, i.e., its eigenmodes [17]. The Koopman operator gives an understanding of the general parameters and model equations that govern the system using only data as input. This is especially useful in biology, where models are not readily available or are simply assumed and could further facilitate understanding of evolutionary adaptive mechanisms as well as their control [53].

Kernel DMD for multiome network integration

While standard DMD is a linear method to identify the underlying forward dynamics F in (1), it cannot fit well to highly non-linear systems as found within molecular interaction networks. Relatively improved DMD algorithms such as extended DMD (eDMD; [39]) or sparse identification of nonlinear dynamics (SINDy; [38]) can disambiguate between linear and non-linear dynamics but do not scale well to highly dimensional systems such as omics datasets. More recently, [37] developed LANDO (Linear and Nonlinear Disambiguation Optimization) that both applies well to non-linear and high-dimensional systems via the use of kernels. Kernels are continuous functions that combine features as inner products within a given Hilbert space H_k where there is a mapping . The true dynamics F (1) can be modelled using a model f(x) by letting

(7)

where ϕ is a candidate model term for every feature in X and ξ is the coefficient that determines how active is each ϕ. For large feature dimensions N, finding suitable terms in (7) becomes computationally expensive and the optimization problem is very hard to solve. The problem (7) can be rewritten in terms of a linear combination of kernels

(8)

where m is the total number of training samples and the weight vectors [37]. The problem now scales over the number of snapshots m instead of the number of features, which makes it more tractable to solve. The full equation of dynamics is

(9)

where g(X) is an eigenfunction (8). We set as the Elastic Net regularization for constraining the weights w as in:

(10)

with α = 1 (penalty coefficient) and ρ = 0.99 (ElasticNet mixing parameter that ensures uniqueness of solution). We also use this type of regularization in subsequent steps. Elastic net regularization has been proven helpful as a constraint in other DMD studies [54] to promote sparseness and was more effective for our dataset compared to just L₁ norm regularization. The value α scales the strength of regularization strengths, with larger values prioritizing sparsity terms preventing overfitting. Values of α below 0.01 lowered the output reconstruction accuracies during the phenocopy of CAM, but did not affect data reconstruction accuracies.

The problem in (8) reduces the parameter search for g to a linear search for weights while combining the original features in X using non-linear kernel functions in k. This method is also suitable for high-dimensional systems, like our dataset, since it scales well with the number of samples and not the feature size. We evaluated the performance of 4 different kernels (linear, gaussian, polynomial and sigmoidal, Table 1) using data reconstruction accuracy (coefficient of determination, R²) to see which one best approximates the dynamics of multiome networks. For each one, we carried out a hyperparameter search by varying their respective parameters (Table 1). The linear kernel is the dot product between features and computes all linear combinations of the feature input space. The gaussian kernel can be interpreted as defining the characteristic length scale of features in x. Thus, concentrations between features that have similar distributions have higher values in the gaussian kernel, while trajectories that are too disparate between them will have almost 0 value. The polynomial kernel computes the degree-d polynomial between features and thus considers quadratic, cubic and higher order interactions between features. This kernel allows for highly non-linear interactions between features that can also be found in Michaelis-Menten dynamics [58]. Finally, we also consider the sigmoidal kernel for inter-omics interactions, which in combination with the linear kernel as a degradation term can approximate Hill function dynamics of the form which are often used to model molecular networks (see [59] for a full derivation and S1 Appendix).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Kernel functions, their respective gradients and hyperparameters.

https://doi.org/10.1371/journal.pcbi.1014029.t001

Within the sigmoidal kernel we also implemented the graph Laplacian L (see Table 1) for integration of a-priori knowledge of molecular pathways in terms of protein-protein (PPI) and protein-metabolite interactions. Similar to FMvPCI [60], which integrates prior knowledge of protein–protein interactions (PPI) via a multiview fusion approach, our method incorporates PPI data obtained from the STRING database and protein-metabolite interaction information from KEGG, allowing for better-informed dynamic representations of molecular networks. As demonstrated by [61] in their work on Link-Based Attributed Graph Clustering, combining structural information and node attributes significantly improves clustering accuracy in complex networks. This approach helps overcome challenges related to the noise and incompleteness of biological interaction data. An adjacency matrix M was constructed for the retained input data after PCA data reduction and a threshold of 0.9 confidence score was used to retain the most significant interactions. The graph Laplacian L was found by

(11)

where D is the degree matrix obtained by setting the sum of incoming edges for each row entry of A in each of diagonal. The Laplacian encodes the local connectivity of data points in the graph. By incorporating it within the dot product of the sigmoidal kernel, the global and local structure of the data is included in the similarity measure [62,63]. The kernel will now be sensitive not only to the raw similarity of feature vectors but also to the relationships encoded in the graph, which capture higher-order dependencies between data points.

Multi-objective function for model weights

For the combination of all omics subsystems into a single linear operator, we concatenate them into one input array where each subscript corresponds to a different subsystem. We apply each kernel to the concatenated multiome and map it forward in time. At the same time, we map inputs to physiological outputs Y in the following multi-objective function:

(12)

where the dash in corresponds to one timestep forward as in (3), X are the multiome inputs and Y are relevant output data that differentiate between strong and weak CAM as defined by common knowledge (see below). w_A are the weights that map the kernel of the multiome inputs X forward in time to X’, while w_C are the weights that make the kernel mapping to the outputs Y. These weights are crucial because they determine how much influence each kernel function has on the optimization, essentially controlling the model’s fit to the data. The weights are learned through approximating Bayesian posterior distributions using a linearized approach (solver ‘CLARABEL’, cvxpy in python) where the objective function is minimized iteratively. At each step, the optimization adjusts the weights in order to reduce the overall error between the predicted values (based on the weighted kernel) and the target values. The representation theorem in kernel learning [64], which states that a nonlinear function can be represented as linear functions in a higher dimensional feature space, ensures that the kernel can be mapped linearly forward in time via the use of these weights.

Solving the multi-objective optimization problem (12) means finding a subset of eigenfunctions of each subsystem X_i (where j is the variable index corresponding to each of the three omics data types, e.g., 1 for transcriptome, 2 for proteome, 3 for metabolome) that are simultaneously required to capture the output dynamics Y. Because we combine both objectives in one equation, the Koopman dynamics, eigenvalues, and eigenmodes are constrained to distinguish explicitly between the two types of photosynthesis captured by differences in Y. To form the concatenated outputs in Y we chose malate, gas exchange measurements H₂O/ CO₂ (mM m^-2 s^-1/ µM m^-2 s^-1), Phosphoenolpyruvate Carboxykinase 1 (PCK1; OG0002610::H1), Phosphoenolpyruvate carboxylase 1 (PEPC1; OG0001285::H1), alpha-glucan phosphorylase 2 (PHS2/PHO1; OG0004358::H2) and pyruvate phosphate dikinase (PPDK; OG0001386::H1) as distinguishing CAM signatures between the two species. PHS2/PHO1 and PPDK are involved in plastidic starch recycling and carbon breakdown pathways, which as mentioned before, were found to contribute to differential evolutionary CAM trajectories in the Clusia species, based on a genome pseudogenization analysis. All outputs in Y are z-scored. By finding suitable weights w_A and w_C we construct our model F(x) (1) by mapping nonlinear interactions within the multiome data X discretely forward in time (i.e., to X’) and to output measurements (Y). The idea is to relate the dynamics learned from the Koopman operator to outputs Y, by evaluating how each eigenmode reconstructs each output (see below in Methods). In this way there is a spatiotemporal link of the original feature space to each output and its relation to CAM photosynthesis via the eigendecomposition of the multiome dynamics. We also use the same types of outputs Y for the phenocopy of the output signature of C. rosea within C. major (see Methods below) showcasing the ability of our control pipeline in shifting C3 output measurements to CAM trends.

Linear Koopman operator from Taylor series expansion

The learned model w_Ak(X_j, X) (12) captures the full forward dynamics but is implicit in nature: it doesn’t provide adequate interpretation of the physical interactions in the system. For this we need to extract and disambiguate the linear from nonlinear dynamics of the model by using a Taylor series expansion [37] around a base state which is taken as the mean of the input states over all features, as in

(13)

The linear operator consists of the partial derivatives of the rate of change around base state which is also the gradient, or the linear Koopman operator. Since the model is composed of linear combinations of kernels the gradient can simply be expressed as

(14)

The respective kernel gradients can be found in Table 1. The Koopman operator A, can be interpreted as a multiome network, where the nodes are the features of the original feature space and edges are the magnitude of rate of change from one node to the other. Thus, we begun with a full kernel model of the forward dynamics and ended with the identification of the linear Koopman operator, which is the final goal of any variant of DMD.

Eigendecomposition of Koopman operator

The SVD decomposition of the kernel gradient leads to the reduced linear Koopman operator by projection of the full Koopman operator A onto the columns of U. If the eigendecomposition of is then

(15)

are the eigenmodes of the system, where W are the learned weights. The eigenmodes (15) of the Koopman operator have a clear physical interpretation where the eigenvectors Ѱ are the spatial modes and represent the temporal components [37]. Stable constant temporal dynamics can be found within eigenmodes that have eigenvalues close to the unit cycle and can correspond in the case of biological systems to maintenance of homeostasis or dynamics that need to be sustained within the diurnal cycle of the plant. Decaying eigenmodes have eigenvalues that lie within the unit cycle (absolute values less than 1) and can be part of pathways that are active in one part of the day and then decay in another. Oscillating dynamics (like the circadian clock) can be aligned with eigenmodes that have imaginary parts in their eigenvalues, and these come in conjugated pairs. The combination of eigenmodes completely describe the dynamics of the linearized system represented by the Koopman operator and can be used to functionally distinguish two dynamic systems in terms of their temporal components. Each eigenmode has a rank of absolute values of each feature in the original feature space and this provides a natural clustering of the network nodes. Features with higher ranks in one mode can contribute more to its type of dynamics than within others. We chose to retain the number of eigenmodes equal to the rank of the kernel’s gradient singular values with a cumulative sum ratio of more than 0.9, which represent the most energetic modes (Fig 2).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Eigenmode decomposition of the two Clusia species.

The eigenvalues and their respective eigenmode amplitudes (same shape) are shown for C. major (top row) and C. rosea (bottom row) from one example iteration. The eigenmodes can completely characterize the decomposed dynamics of the system. Eigenvalues within the unit cycle describe stable eigenmodes with decaying dynamics, eigenvalues near the unit cycle are for sustainable dynamics and eigenvalues with imaginary parts for oscillatory dynamics. Each eigenmode has a feature rank which describes its qualitative nature. This is an example from one iteration of the kernel-DMD algorithm which introduces some variability in the eigenmodes due to stochastic initialization and additive noise added at each iteration. Overall eigenvalues and eigenmode amplitudes are expected to follow trends as shown in this figure.

https://doi.org/10.1371/journal.pcbi.1014029.g002

We associated the qualitative nature of each eigenmode using the modal reconstruction accuracies of CAM-related outputs (Fig 3). In order to make valid conclusions on the qualitative nature of eigenmodes, we chose to average results over many iterations selecting those eigenmodes that predicted CO₂ gas exchange outputs well (R² score > 0.3), since stomatal conductance was the main physiological difference between C3 and CAM dynamics in C. major and C. rosea respectively. In particular, for each iteration the weighted graph G of the selected eigenmodes, for which only the first 60 features with highest ranking absolute values were kept, was calculated as an outer product [65]

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Koopman and eigenmode phenotype reconstructions.

The fit of the kernel was evaluated based on Koopman and eigenmode reconstruction accuracies (R² score) of the phenotypes. The eigenmodes that predicted each output well distinguish the two species based on their feature ranks and temporal dynamics. In this way the whole multiome network can be decomposed into its temporal constituents which characterize different system outputs that relate to the different forms of photosynthesis between the two species. Due to the variability in the eigendecomposition introduced by stochastic initiation and resampling in kernel-DMD, the figure shows an example of how each eigenmode relates to the outputs. This can slightly change between iterations and therefore we do not elaborate on the biological significance of modal reconstruction accuracies from this particular iteration.

https://doi.org/10.1371/journal.pcbi.1014029.g003

(16)

keeping edges with absolute weight above the upper quantile. The retained edges were added after each iteration to a consensus graph and edges with frequency above 0.5 were kept (S1 and S2 Tables).

Extracting the control input matrix from Taylor series residuals

Since CAM is a photosynthetic mode with higher water use efficiency, one aim was the identification of optimal control that shifts dynamics from C3  levels (as in C. major) to strong CAM (as in C. rosea) dynamics (Fig 1B). To this end, the nonlinear higher order terms from the Taylor series expansion (13) act as a nonlinear forcing to the linear operator A and can be used to identify control matrices. N is found by

(17)

which is the nonlinear residual term after subtracting the base state (input mean) and linear (gradient) dynamics from the whole state dynamics F(x) [37]. In order to extract meaningful information from N, it was linearly decomposed by SVD yielding left singular vectors that span the image (output or column space) of the nonlinear effects. These vectors represent the dominant directions or modes of nonlinear forcing in state space where it is projected or acts on. Multiplying the left singular vectors by the singular values after truncation gives

(18)

where B is the control matrix (Fig 1B). Scaling by singular values amplifies the effect of the dominant nonlinear modes according to their singular values. The cut-off value for truncating Σ was chosen as the number of singular values whose cumulative sum exceeded 90% of the total sum of all singular values, explaining more ‘energy’ in the nonlinear part. Conversely, the right singular vectors after SVD of N(x), give time coefficients of the contribution of each nonlinear mode in U across time.

Data reduction using Hankel singular values

Hankel Singular Values were used for feature selection in order to reduce the system to a smaller subspace in terms of dynamics. This required the calculation of the controllability and observability Gramians that make use of the linear operators A, B and C derived in previous steps. The controllability Gramian W_c and observability Gramian W_o are expressed as infinite integrals [66]:

(19)

W_c captures how easily each internal state is reached by the inputs [67] and W_o, j how each internal state affects each output j (j is part of Y) [68]. They are solutions to the Lyapunov equations:

(20)

The product between the two Gramians gives the Hankel matrix H [69]

(21)

which reflects each state’s combined observability and controllability, i.e., how much each state contributes to outputs but also how easily it is controlled by the inputs. The Hankel singular values (HSVs) σ_i [70,71] are defined as

(22)

where λ_i are the eigenvalues of H, quantify the energy of each state or how important it is in terms of dynamics. HSVs decay rapidly in large-scale systems [72], which facilitates low-rank approximations by taking threshold values for truncation at the point of decay. The features in the multiome data corresponding to large HSVs were kept and system reidentification for all linear matrices (A, B and C) was carried out in the reduced subspace using the optimal parameters chosen by the hyperparameter search of the multi-objective function (12).

Koopman matrix alignment between species

Alignment of the new reduced Koopman matrices was carried out before applying control by using a transformation matrix T to align the Koopman matrix of C. major (A₁) to that of C. rosea (A₂) as in

(23)

Transformation matrix T aligns both Koopman matrix A₁ to A₂ akin to Procrustes analysis of vector fields [73] but each matrix is also multiplied by their respective output matrices C which means T jointly minimizes the distance between internal and output dynamics of the two systems. This allows the mapping of the internal state of one system to match that of another while at the same time being projected onto their respective outputs. The new Koopman matrix for C. major

(24)

is used as input to linear MPC. The Koopman operator of C. major has to firstly be projected in the right subspace (by transformation matrix T) for correct alignment of the product of Koopman and output matrices between the two species before optimal control can find the best control input solutions for proper phenocopying.

Linear model predictive control

MPC is implemented as in [74], where C. major inputs are optimized to match the outputs of C. rosea for correct phenocopying. The cost function is defined as

(25)

where z are the outputs of C. major, z_ref the outputs of C. rosea, Q, R₁ and R₂ the diagonal output, control input and input penalty matrices respectively, and t the number of timepoints over which to iterate, with a one-timestep prediction at each iteration (assuming perfect outcome). The objective is to minimize the cost function J by finding optimal inputs u_i to minimize the difference between the outputs of the two systems. The input-output equations are placed as constraints to the objective function as in:

(26)

where K is a gain matrix to be found, x the C. major inputs to be optimized, x_ref the known C. rosea inputs, the Koopman operator, C the output operator and B the control operator. The effectiveness of input control is evaluated based on reconstruction accuracies (R²) of output values of C. rosea. The gain matrix K was found by solving the discrete-time algebraic Riccati equation [75]

(27)

to find P with R₁, R₂ and Q are the weight matrices used in (25) and are vital for optimizing the control inputs and minimizing cost J. R₁ and R₂ are identity matrices and Q has a weight of 5¹⁰ in its diagonals putting higher weight on matching outputs. P is then used as input to

(28)

to find K.

Sensitivity gain matrix

The importance of each feature in linear MPC was evaluated based on the sensitivity gain matrix [76] S found by

(29)

of which the mean for every feature was taken after many iterations. Sensitivity values give a measure of how small changes in the input influence the state dynamics. The full derivation of S (29) can be found in S1 Appendix.

Code implementation

We use the cvxpy package in Python for the multi-objective optimizations with the ‘CLARABEL’ solver as it produced the highest fit accuracy scores for our quadratic objective functions. We parallelize the code using loky backend to iterate the algorithm 100 times, reducing computation time from hours to a minute. Multiple iterations were required to validate robustness of results across repetitions as well as the calculation of consensus eigenmodes and mean sensitivity values. A leave-3-out cross validation was repeated 10 times to validate the main model, by excluding the last 3 timepoints in the test set. The following versions of python and packages were used to create a virtual environment: python 3.13.7, pandas 2.3.3, numpy 2.2.6, scipy 1.16.2, cvxpy 1.6.0, scikit-learn 1.7.2, joblib 1.5.2, matplotlib 3.9.4, networkx 2.5, seaborn 0.13.2.

Part I: System identification

Kernel choice determines the quality of nonlinear multiome integration.

We evaluated four kernel families (linear, Gaussian, polynomial, and sigmoidal) within the multi-objective optimization framework to determine which best captured the nonlinear dynamics of the concatenated transcriptome, proteome, and metabolome datasets using kernel DMD (see Methods). For the data and each output, we evaluated the reconstruction accuracy (R² score) using the full Koopman operator in linear space which was derived by the using the learned weights of the kernel in (12) and the kernel gradient (14). The kernel evaluations and the hyperparameter search results are shown in Table 2. The accuracy of the sigmoidal kernels remains consistently high (around 0.7) for both species regardless of the coefficient of the sigmoid (relative to the linear decay) term. The accuracy decreases when the coefficient γ increases combined with a higher sigmoid kernel term. As the term represents the interaction term between inputs, when it is larger it tends to overfit noise and outliers and does not generalize over the whole input space. It thus creates a steeper threshold boundary that might create vanishing gradients used for the linearization step (13, 14) that does not fall within the base state (mean) of the inputs. The robustness of the sigmoidal kernel was also very high with <0.05 standard deviations of R² score. Comparing with the other kernels, the sigmoidal kernel performed much better than the gaussian or polynomial kernels, justifying its choice for fitting to this data. Where usually the gaussian kernel almost always fits well to any dataset due to its good generalizability [37,77,78], here it gave much lower accuracies. It should be noted that the success of the sigmoidal kernel is attributed to its second linear degradation part x^Tx which is also used on its own as part of the linear kernel (hence the high accuracy of the latter). If this term is added to any other kernel along with its corresponding derivative term in the gradient, then the linear model of the respective kernel succeeds in reconstructing the data and outputs with high accuracies regardless of the first part of the kernel (whether it be gaussian, polynomial or sigmoidal). We choose to proceed with the “Hill-function”-like kernel to showcase that tailored kernels can be combined to produce representations of functions applied to molecular systems and lead to conclusions aligned with the literature in relation to distinctions between C3 and CAM photosynthesis as shall be presented later in the results. Each kernel was also evaluated based on the ability of the linear Koopman operator and its eigenmodes to reconstruct each output Y (12). Output reconstructions of the sigmoidal kernel using the linear Koopman operator were moderate to high (>0.6 R² Fig 3) for all outputs and both species. To avoid risk of over-fitting due to high-dimensionality of the data, we repeated a leave-3-out cross validation 10 times by excluding the last 3 timepoints in the test set. Overall R² accuracy in the test set was high (0.7 for data, average 0.7 for outputs), which indicated that the model trained on the training data generalized well and did not overfit.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Kernel function evaluations.

https://doi.org/10.1371/journal.pcbi.1014029.t002

Eigenmode decomposition reveals latent temporal modules.

The eigendecomposition of the linear Koopman operator (see Methods) results in the decomposition of dynamics into distinct temporal components, or eigenmodes, each representing a dynamical module contributing to the evolution of the system over the diurnal cycle. Each eigenmode and its temporal dynamic (Fig 2) can be linked to each output via reconstruction accuracy (R² score; Fig 3). Modal reconstructions provide an indirect link between features (through the rank of the feature’s absolute value in the eigenmode) and how they best predict each output across the timeseries. Thus, a direct comparison can be made between the underlying multiome pathways that differentially predict ”C3” and strong CAM outputs of the two species. Figs 2 and 3 show the eigenvalues, eigenmode amplitudes, and mode-specific reconstruction accuracies from one representative iteration of the kernel-DMD optimization. Because kernel-based system identification involves stochastic initialization and resampling, individual eigenmodes vary slightly between iterations. Therefore, these figures serve to illustrate the spectral structure and qualitative modal types (stable, decaying, oscillatory). To extract biologically meaningful information, we do not interpret individual modes from a single iteration. Instead, we compute a consensus across 100 iterations, retaining only those eigenmodes that consistently predict CO₂ gas-exchange dynamics (R² > 0.3). The consensus eigenmode topologies (Fig 4 and S1–S2 Tables) provide the stable spatial patterns that persist despite stochastic variability and are biologically interpretable. These are the modes used in all pathway-level and biological interpretations discussed below. Importantly, the eigenmodes emerging from the Koopman operator naturally contain a mixture of transcript, protein, and metabolite features. This occurs because the kernel and graph Laplacian couple features across omics layers, enabling the model to learn their coordinated temporal behavior rather than treating each layer independently. The resulting consensus eigenmode clusters (Fig 4) therefore represent functional modules spanning transcriptional regulation, enzymatic activity, and metabolic flux, consistent with the cross-omic coordination required for CAM and C3  photosynthesis.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Consensus eigenmode topology for each species.

The networks shown represent the consensus topology of eigenmodes that consistently predicted CO₂ gas-exchange outputs across 100 kernel-DMD iterations (R² > 0.3). Nodes correspond to multiome features (transcripts, proteins, metabolites), coloured according to whether they are transcript [12], protein (blue) or metabolite (orange). Edges represent feature pairs that co-occurred within the same eigenmode with frequency >0.5 across iterations, reflecting stable dynamical associations. (A) C. rosea. The consensus clusters highlight enhanced flux through glycolysis, starch remobilization, and key CAM-associated enzymes (e.g., PEPC, PPCK1, PHO1). Features from the TCA cycle (ACO, CAC3) and sugar interconversion pathways (RFS1, BGAL1) also appear as highly ranked and strongly connected nodes. This topology reflects a night-time metabolic program supporting PEP regeneration, organic-acid accumulation, stomatal opening at night, and CAM-type carbon assimilation. Together, these consensus graphs reveal stable dynamical modules that differentiate C3  (C. major) and strong CAM (C. rosea) metabolism, independently of stochastic variation across individual DMD iterations. (B) C. major. The dominant consensus cluster is enriched for photorespiratory enzymes (e.g., RBCS, GDCSH, GDCSP, CTIMC), Calvin-cycle intermediates, and glyoxylate metabolism. This topology indicates that C. major dynamics are strongly shaped by daytime RuBisCO oxygenase activity, carbon recycling through the photorespiratory pathway, and metabolic routing toward 3-PGA production.

https://doi.org/10.1371/journal.pcbi.1014029.g004

Part II: System control and phenocopy of CAM dynamics

Dynamic feature selection via Hankel singular values.

Due to the high dimensionality of the system, a smaller state space representation of dynamics was sought using dynamic information coming from the Koopman (A) and control input (B) matrices. Feature selection was carried out according to observability and controllability metrics, a method largely developed by Willems [84] and later used by others [85]. Model reduction was carried out by taking the first 50 indices of HSV values after ranking them in descending order for all outputs. The number of indices was selected as a balance between good reconstruction accuracy of C. rosea outputs after linear Model Predictive Control (MPC, see Methods) in the selected reduced feature space of C. major and minimal number of features kept. The final reduced features spaces included unique features across both C. rosea and C. major according to this ranking for each output. The top features selected over 100 iterations include glycolysis enzymes such as GAPC, FBA, GAPA and FBP as well as Calvin cycle intermediates such as glycine dehydrogenase (GDCSP) and transcription factors GATA1 [86] and PCF1 [87], involved in regulating gene expression in response to nutrient availability and light intensity, helping regulate stomatal movement and stress responses. The robustness of HSV rank as a feature selection method is shown in Fig 5A which shows the features retained after data reduction and their frequency over 100 iterations. Thus, a solid observability-controllability metric (the HSV rank) was used to select features which play dual roles: they strongly shape system outputs and their levels can be effectively influenced under control. This data reduction step leads to efficient application of control in a smaller subspace.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. System Control output results.

A) Robustness of data reduction via HSV feature selection across iterations. Data reduction was carried out by truncating Hankel Singular Values below a threshold. The frequency of each retained feature after 100 iterations is shown on the x-axis for C. major and C. rosea. B) Phenocopy of the C. rosea outputs in C. major. Prediction accuracy (R² score) C. rosea outputs in C. major after MPC optimization of control inputs. Relatively high accuracies are achieved after implementing the control strategy to shift the C. major output distributions to match those of C. rosea, highlighting the success of the workflow. C) Sensitivity gain matrix values. Positive entries [12] indicate features with higher positive feedback input in C. major and negative entries (blue) features with negative feedback input. Glycolytic enzymes (PFK3, GAPN, FBA2 and SPS1) mostly have positive feedback and photorespiratory enzymes (RBCS, GDCSH, GDCSP) have negative feedback. This agrees with the widely known role of glycolysis in maintaining CAM levels and of photorespiration being active in C3 types of photosynthesis.

https://doi.org/10.1371/journal.pcbi.1014029.g005