Source-linked AI summary
Data-Based Clustering and Control of Similar Biological Systems
Peilin Zhang, Antonis Papachristodoulou, Idris Kempf
TL;DR
Large heterogeneous biological populations are difficult to control because conventional strategies require substantial computation for each system. The paper clusters systems using data-driven control-relevant similarity, assigns one leader controller per cluster, and derives well-posedness conditions. In simulation, clustering 100 systems into two groups reduced controller synthesis effort by 98% while maintaining consistent performance across clusters.
Problem
Per-system control computation limits scalable real-time control of large heterogeneous biological populations.
Method
The paper clusters systems using data-driven control-relevant similarity and applies a hierarchical leader–follower architecture with one controller per cluster.
Results
Reducing 100 systems to two clusters cut required controllers from 100 to 2, a 98% reduction in control synthesis effort, while performance remained consistent across clusters.
Takeaways & Limitations
Similarity-based grouping can enable scalable, parallel control of heterogeneous systems while incurring only small closed-loop performance degradation in the studied setting.
Abstract
from arXiv · showhide
Cybergenetic control of gene expression enables applications in synthetic biology, drug development, and biomanufacturing. Microfluidic platforms allow the parallel control of large cell populations. However, the resulting computational burden and intrinsic biological heterogeneity limit the scalability of conventional control strategies. In this work, we propose a similarity-based framework to reduce the computational requirement of controlling large numbers of dynamical systems. Building on existing data-driven methods for quantifying control-relevant similarity from input-output data, we cluster systems with similar dynamics without requiring explicit system identification. Based on this grouping, we develop a hierarchical leader-follower control architecture, where a single controller is designed for each cluster and applied to all members. This significantly reduces the number of control problems that need to be solved. Furthermore, we analyse the closed-loop behaviour within clusters and develop data-driven conditions under which the clustered closed-loop systems remain well-posed. The proposed approach is demonstrated in simulations of gene expression dynamics, showing that similarity-based grouping enables scalable and reliable control of heterogeneous biological systems.
I. INTRODUCTION
The paper targets scalable control of heterogeneous biological systems, where cell-level variability and per-system computation limit conventional approaches. It proposes data-driven similarity clustering with hierarchical leader–follower control to reduce repeated controller design.
- Thousands of parallel optimisation or control problems can make real-time control computationally limiting as population size grows.
- The framework clusters systems using control-relevant, data-driven similarity rather than heuristic similarity measures.
- A hierarchical leader–follower architecture performs computationally intensive operations only for cluster leaders and reuses them across followers.
- The approach combines similarity-aware clustering with data-based linear quadratic control for simulated biological systems.
II. BACKGROUND AND PRELIMINARIES
The paper represents dynamical systems through finite-horizon input–output behaviours and compares them using data-driven subspace geometry. These constructions support similarity assessment without requiring explicit system models.
- The data-driven representation uses restricted behaviours containing length-L trajectories, including arbitrary initial conditions.
- A finite-horizon graph isolates zero-state input–output trajectories from the broader restricted behaviour.
- Willems’ Fundamental Lemma permits restricted behaviours to be obtained directly from sufficiently informative input–output trajectories.
- The L-gap compares restricted behaviours through the operator-norm distance between their orthogonal projectors.
- For equal-dimensional restricted behaviours, the L-gap is the sine of their largest principal angle and can be computed from subspace bases.
A. Clustering
The paper develops data-driven graph comparisons and feedback conditions for transferring a controller from one dynamical system to another. These results provide the theoretical basis for clustering systems with reusable controllers.
- A. Clustering: A controller well-posed for one system remains well-posed for another when their graph L-gap satisfies the derived finite-horizon condition.
- A. Clustering: The proposed graph-based construction enables controller-transfer analysis directly from data rather than explicit system identification.
- A. Clustering: Finite-horizon feedback well-posedness is characterized by the direct-sum decomposition of the plant graph and inverse controller graph.
- A. Clustering: The graph L-gap compares zero-state input–output maps, unlike the L-gap for behaviours that include arbitrary initial conditions.
- A. Clustering: The finite-horizon graph can be extracted from input–output data by imposing zero input–output history and taking the image of the resulting Hankel-data matrix.
B. Performance
The paper bounds finite-horizon trajectory and performance differences when one controller is shared between a leader and a similar follower. These bounds support well-posed clustered control and quantify the cost of reusing controllers across systems.
- B. Performance: The closed-loop trajectory comparison decomposes each exogenous input into system and controller-graph components before bounding their difference.Projection onto the controller graph separates the relevant trajectory terms.
- B. Performance: Well-posedness of both leader and follower interconnections follows when the relevant graph gap is below one.The restricted controller projection is bijective under this condition.
- B. Performance: The trajectory mismatch between leader and follower is bounded by ηw|e|0 under the theorem’s assumptions.This bound provides the basis for assessing how similarity affects closed-loop behavior.
- B. Performance: The performance difference satisfies |J(w2) − J(w1)| ≤ ηJ|e|0^2 when both systems receive the same exogenous input.The coefficient ηJ depends on the trajectory bound and the performance weighting matrix.
IV. CASE STUDY: CYBERGENETIC CONTROL
The case study targets cybergenetic control in microfluidic bacterial populations, where many cells must be controlled at experimentally relevant sampling rates. The evaluation uses 100 systems on a standard laptop to examine whether clustering reduces controller-synthesis burden.
- IV. CASE STUDY: CYBERGENETIC CONTROL: Microfluidic cybergenetic applications can involve populations of up to 10^6 cells, creating a substantial computational burden for control at relevant sampling rates.This motivates reducing the number of controller synthesis problems.
- IV. CASE STUDY: CYBERGENETIC CONTROL: The study evaluates the framework on N = 100 systems using a standard laptop with an Intel Core i7 processor and 8 GB RAM.The systems represent bacterial populations controlled in microfluidic platforms.
A. Biological Model
The biological model represents an E. coli strain expressing a GFP reporter regulated by the optogenetic CcaS/CcaR system. The analysis linearizes the model around a nominal operating point and assumes measurable mRNA and GFP concentrations for state feedback.
- A. Biological Model: The model describes mRNA and GFP dynamics in an E. coli strain controlled through the optogenetic CcaS/CcaR system.The state variables are m(t), the mRNA concentration, and M(t), the GFP concentration.
- A. Biological Model: The system is linearized at u⋆ = Kf, with nominal mRNA and GFP concentrations determined by the transcription, translation, and degradation parameters.The operating point is specified by u⋆= Kf, m⋆= (km + f(u⋆))/γm, and M⋆= kMm⋆/γM.
- A. Biological Model: The analysis assumes both m(t) and M(t) are measurable because subsequent controllers use state feedback.The discretized model uses zero-order hold with sample time Ts = 10 min.
B. Data-generation
The case study models biological heterogeneity by varying selected kinetic parameters and constructs graph subspaces directly from input–output data. Persistent excitation, Hankel matrices, and an SVD provide the data-driven representation used for clustering.
- B. Data-generation: The kinetic parameters γm, γM, km, and kM are sampled independently and uniformly from 0.8 to 1.2 times their nominal values.The parameters α, Kf, and n remain fixed, and the formulation assumes linear dynamics around a nominal operating point.
- B. Data-generation: For each system, noise-free input–output data form partitioned Hankel matrices with L = 10 and Tini = n = 2.The input is persistently exciting of order Tini + L + n.
- B. Data-generation: An SVD of Z = col(Uf, Yf)Np yields an orthonormal basis for the graph subspace used in the data-driven construction.Np spans ker(col(Up, Yp)), so Z spans GL(G).
C. Similarity-Based Clustering
The framework clusters systems using control-relevant graph L-gap distances and assigns one leader to each cluster. Farthest-point-first clustering promotes coverage of behavioral diversity while avoiding groups with dissimilar dynamics.
- Distance-based clustering: The distance matrix contains pairwise graph L-gap distances, and systems are partitioned into K nonempty, disjoint clusters.
- Leader selection: Each cluster designates one leader, and a single controller designed for that leader is applied to all cluster members.The clustering objective is to group systems with small pairwise graph L-gap, with Thm. 1 providing a sufficient distance for preserving well-posedness and Prop. 3 relating smaller distances to lower performance-degradation bounds.
- Clustering strategy: Farthest-point-first clustering selects maximally separated initial members to cover behavioral diversity and avoid grouping dissimilar systems.Classical K-means may be unsuitable because the graph L-gap is not a Euclidean distance.
- Simulation setup: For 100 bacterial systems, two clusters satisfy the well-posedness bound, while five clusters produce more closely matched behaviors.The two- and five-cluster configurations are compared using the L-gap to each cluster leader.
D. Leader–Follower Control Architecture
The leader–follower architecture designs one data-driven controller per cluster and reuses it across that cluster’s systems. The simulations indicate reliable performance, bounded leader–follower cost differences, and substantial computational savings.
- Architecture: A controller is designed solely for a representative leader in each cluster and then applied to all systems within that cluster.The controller uses a data-driven linear quadratic regulator formulation.
- Computational cost: Controller synthesis scales as O(NT^3n^3) for N individually controlled systems but reduces to O(KT^3n^3) when grouped into K ≪ N clusters.The dominant computational cost is the semidefinite-program solve.
- Performance and scalability: Clustered closed-loop systems remain well-posed, and performance degradation relative to individually designed control does not change drastically across systems.Smaller clusters further reduce degradation, creating a trade-off between cluster size and performance.
- Performance and scalability: Reducing 100 systems to two clusters cuts the number of required controllers by 98% while maintaining consistent performance across each cluster.The runtime remains shorter than the individual-controller baseline for up to at least 100 systems, including clustering overhead.
- Theoretical validation: For the two-cluster case, all observed leader–follower cost differences lie within the theoretical bounds from Prop. 3.The comparison uses a disturbance step and the cost difference defined in (7).
V. CONCLUSION
The paper presents data-driven control-relevant clustering as a way to reuse controllers across heterogeneous dynamical systems while preserving closed-loop guarantees. Its preliminary evaluation shows small performance degradation and reduced computational requirements, but the current formulation remains constrained by pairwise clustering costs and LTI assumptions.
- Conclusion: The framework efficiently clusters many systems into a few groups with only small closed-loop performance degradation in the evaluated setting.The study uses data-driven LQR and a sufficient data-driven condition for closed-loop well-posedness.
- Computational implications: Clustering N systems into K ≪ N groups reduces online controller-design and implementation complexity, although pairwise clustering introduces poorly scaling overhead.The benefits are expected to outweigh this overhead for more demanding control algorithms such as data-based predictive control.
- Limitations and future work: The current framework computes all pairwise gaps, whose quadratic scaling motivates adaptive or hierarchical clustering strategies for future work.Online data updates can also make the L-gap time-varying and require repeated Hankel-matrix factorization.