Data-driven analysis has seen explosive growth with widespread availability of data and unprecedented computational power. Time-series data is of particular interest from a systems theory perspective which seeks to predict behaviour of involved entities and uncover unforeseen events. While system identification has been vastly successful in constructing models for linear systems, the Koopman operator framework has gained popularity due to its relations to nonlinear systems. The Koopman representation's linearity in functions of state and beyond local validity has been exploited by the dynamic mode decomposition (DMD) class of algorithms for their ease of identifying system models from data.

We study the effect of some attributes of data -- number of time-points, power-spectrum, periodicity, sampling, and number of trajectories -- on the results of Extended-DMD (EDMD). After a brief overview of the Koopman framework, we derive EDMD as a projection onto a basis without assumptions on desired functions lying in the span of that basis. We apply persistence of excitation (PE) to the Koopman framework and prove necessary and sufficient condition on the data for accuracy of EDMD approximation of the Koopman operator for nonlinear systems - using the basis at the data-points. Findings from this motivated us to pursue an order reduction algorithm for EDMD that has been known to suffer from the curse of dimensionality. We prove properties of matrices introduced for EDMD to list conditions and give solutions compiled into Reduced Order EDMD (RODMD), an algorithm that modifies the EDMD basis such that it minimizes the error in approximating desired functions while also keeping the dimension at its lowest. Alongside, we use results on convergence of EDMD estimation to the Koopman operator in literature to compare sampling characteristics of data used. For a certain number of data-points, we prove quantitative convergence guarantee for transient behaviour in terms of number of trajectories that the data is spread across. We use these derivations and proofs to show the rather basic relationship between EDMD and system identification, and use it to show the PE condition on data for uncontrolled systems as necessary for estimation of the Koopman operator.

We also present an application of dynamical systems theory to artificial neural networks to give an example of the interdisciplinary applicability of the Koopman framework. Using a dynamical system perspective of the optimization of neural networks, we prove the existence of explicit representation of some parameters of the neural network in terms of the other parameters and data, thereby paving way for one-shot learning through DMD algorithms.

Synthetic biology has the potential to transform numerous aspects of our human existence, such as the manner in which we combat disease, cultivate food, and produce goods. However, the widespread use of engineered organisms in industrial and medical settings is hindered in part by i) the development of biological tools and parts in model organisms that are not guaranteed to function in application-relevant organisms, and ii) the lack of computational models to assess the robustness of genetic circuit designs and predict transcriptional changes due to the introduction of foreign DNA. Addressing these two challenges is made difficult due to the high-dimensional nature of gene regulatory networks. In this thesis, we address these challenges through a combination of large volumes of data and mathematically-principled approaches.

A major challenge in biotechnology and biomanufacturing is the identification of a set of biomarkers for perturbations and metabolites of interest. In Chapter 2, we develop a data-driven, transcriptome-wide approach to rank perturbation-inducible genes from time-series RNA sequencing data for the discovery of analyte-responsive promoters. This provides a set of biomarkers that act as a proxy for the transcriptional state referred to as cell state. We construct low-dimensional models of gene expression dynamics and rank genes by their ability to capture the perturbation-specific cell state using a novel observability analysis. Providing an optimal selection of reporters, observability analysis identified 15 candidate biosensors for a pesticide in a non-model organism, whose collective response is greater than the sum of its parts. The engineered host cell, a living malathion sensor, can be optimized for use in environmental diagnostics while the developed machine learning tool can be applied to discover perturbation-inducible gene expression systems in the compendium of host organisms.

While observability analysis provides optimal selection of reporters to construct biosensors from, the engineering of microbes to transcribe and translate foreign DNA that comprise the biosensors (or any other synthetic genetic circuit) results in an unintended influence on the host transcriptome the consequences of which can be fatal to the engineered cell. In Chapter 3, we develop structured dynamic mode decomposition (sDMD), comprising of compositional Koopman operators which model the induced transcriptional changes of the host transcriptome by exogenous genes. We consider an experimental example, using high-throughput RNA sequencing measurements collected from wild-type \textit{E. coli}, single gate components transformed in \textit{E. coli}, and a NAND circuit composed from individual gates in \textit{E. coli}, to explore how compositional Koopman models encode increasing circuit interference on the native \textit{E. coli} transcriptome. From this dataset, sDMD can both recover known regulatory biology through recapitulation of known sugar utilization hierarchies and predict new regulatory mechanisms induced by the transcription and translation of foreign DNA.

While useful for modeling circuit-host impact, dynamic mode decomposition has significant drawbacks, the most important being the reliance on uniformly sampled data in time, a feature rarely satisfied by high-throughput biological measurements. This results in subpar usage of transcriptional data for inference of circuit-host impact. In Chapter 4, we develop a deep learning approach to disentangle the factors causing the transcriptional changes on a gene-by-gene basis. The model architecture is inspired by a latent process model in which we treat the latent gene expression and latent perturbations as unobserved processes. We show that we are able to recover known regulatory biology as well as discover new regulatory mechanisms which lead to unintended consequences. Specifically, our circuit-host impact model infers that engineering microbes are more resistant to $\beta$-lactam antibiotics than their wild-type counterparts. This hypothesis is confirmed in gram-negative bacteria.

Nonlinear systems can be decomposed into observable and unobservable subsystems in theory, but achieving this decomposition in a data-driven framework is challenging. Koopman operators enable us to embed nonlinear dynamical systems in high-dimensional function spaces. In this talk, we will explore how the observable decomposition of linear Koopman models relates to the observable decomposition of nonlinear systems and show how this decomposition can be achieved in a data-driven setting.

In a model biological soil bacterium, Pseudomonas putida, we use a deep neural network approach to learn Koopman operator representations to model the gene expression-phenotype dynamics. Using Koopman observable decomposition, we identified 15 genes out of 5564 genes in Pseudomonas putida, which impact the growth phenotype of the bacterium in soil simulant media conditions. Using CRISPRi, we construct synthetic strains that suppress the expression of these target genes and show that 80% of the gene targets have the predicted impact on the fitness of the bacterium Pseudomonas putida.

Our results provide a novel framework called Koopman observable decomposition, which functions as a machine learning tool for detecting critical states that generate desired outcomes in complex, high-dimensional nonlinear dynamical systems.