Scalable nonlinear programming framework for parameter estimation in dynamic biological system models

Recurrent neural networks enable design of multifunctional synthetic human gut microbiome dynamics

Predicting the dynamics and functions of microbiomes constructed from the bottom-up is a key challenge in exploiting them to our benefit. Current models based on ecological theory fail to capture complex community behaviors due to higher order interactions, do not scale well with increasing complexity and in considering multiple functions. We develop and apply a long short-term memory (LSTM) framework to advance our understanding of community assembly and health-relevant metabolite production using a synthetic human gut community. A mainstay of recurrent neural networks, the LSTM learns a high dimensional data-driven non-linear dynamical system model. We show that the LSTM model can outperform the widely used generalized Lotka-Volterra model based on ecological theory. We build methods to decipher microbe-microbe and microbe-metabolite interactions from an otherwise black-box model. These methods highlight that Actinobacteria, Firmicutes and Proteobacteria are significant drivers of metabolite production whereas Bacteroides shape community dynamics. We use the LSTM model to navigate a large multidimensional functional landscape to design communities with unique health-relevant metabolite profiles and temporal behaviors. In sum, the accuracy of the LSTM model can be exploited for experimental planning and to guide the design of synthetic microbiomes with target dynamic functions.

Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.

Isotope-assisted metabolic flux analysis (MFA) is a computationally intensive parameter estimation problem. Isotopically nonstationary MFA (inst-MFA) represents the most computationally burdensome MFA application. We present the formulation of the steady-state and inst-MFA problems as equality-constrained nonlinear programs (NLPs), solved by a state-of-the-art solver implemented in an algebraic modeling language. We show that this formulation leads to robust convergence properties compared to traditional approaches, particularly for inst-MFA. We developed a software, eiFlux that uses the NLP formulation to perform both steady-state and inst-MFA. We demonstrate the application of eiFlux on several examples, including a genome-scale inst-MFA model, and show that it has robust optimal convergence even when started from a very poor initial guess for the parameters. eiFlux is implemented using the Python programming language and the General Algebraic Modeling System (GAMS), using the CONOPT solver. eiFlux is available upon request, pending institutional approval, and is free for academic use.

Kinetics of Asian and African Zika virus lineages over single-cycle and multi-cycle growth in culture: Gene expression, cell killing, virus production, and mathematical modeling

Since 2014, an Asian lineage of Zika virus has caused outbreaks, and it has been associated with neurological disorders in adults and congenital defects in newborns. The resulting threat of the Zika virus to human health has prompted the development of new vaccines, which have yet to be approved for human use. Vaccines based on the attenuated or chemically inactivated virus will require large-scale production of the intact virus to meet potential global demands. Intact viruses are produced by infecting cultures of susceptible cells, a dynamic process that spans from hours to days and has yet to be optimized. Here, we infected Vero cells adhesively cultured in well-plates with two Zika virus strains: a recently isolated strain from the Asian lineage, and a cell-culture-adapted strain from the African lineage. At different time points post-infection, virus particles in the supernatant were quantified; further, microscopy images were used to quantify cell density and the proportion of cells expressing viral protein. These measurements were performed across multiple replicate samples of one-step infections every four hours over 60 h and for multi-step infections every four to 24 h over 144 h, generating a rich data set. For each set of data, mathematical models were developed to estimate parameters associated with cell infection and virus production. The African-lineage strain was found to produce a 14-fold higher yield than the Asian-lineage strain in one-step growth and a sevenfold higher titer in multi-step growth, suggesting a benefit of cell-culture adaptation for developing a vaccine strain. We found that image-based measurements were critical for discriminating among different models, and different parameters for the two strains could account for the experimentally observed differences. An exponential-distributed delay model performed best in accounting for multi-step infection of the Asian strain, and it highlighted the significant sensitivity of virus titer to the rate of viral degradation, with implications for optimization of vaccine production. More broadly, this study highlights how image-based measurements can contribute to the discrimination of virus-culture models for the optimal production of inactivated and attenuated whole-virus vaccines.

Design of synthetic human gut microbiome assembly and butyrate production

The capability to design microbiomes with predictable functions would enable new technologies for applications in health, agriculture, and bioprocessing. Towards this goal, we develop a model-guided approach to design synthetic human gut microbiomes for production of the health-relevant metabolite butyrate. Our data-driven model quantifies microbial interactions impacting growth and butyrate production separately, providing key insights into ecological mechanisms driving butyrate production. We use our model to explore a vast community design space using a design-test-learn cycle to identify high butyrate-producing communities. Our model can accurately predict community assembly and butyrate production across a wide range of species richness. Guided by the model, we identify constraints on butyrate production by high species richness and key molecular factors driving butyrate production, including hydrogen sulfide, environmental pH, and resource competition. In sum, our model-guided approach provides a flexible and generalizable framework for understanding and accurately predicting community assembly and metabolic functions.

Developing political-ecological theory: The need for many-task computing

Models of political-ecological systems can inform policies for managing ecosystems that contain endangered species. To increase the credibility of these models, massive computation is needed to statistically estimate the model’s parameters, compute confidence intervals for these parameters, determine the model’s prediction error rate, and assess its sensitivity to parameter misspecification. To meet this statistical and computational challenge, this article delivers statistical algorithms and a method for constructing ecosystem management plans that are coded as distributed computing applications. These applications can run on cluster computers, the cloud, or a collection of in-house workstations. This downloadable code is used to address the challenge of conserving the East African cheetah (Acinonyx jubatus). This demonstration means that the new standard of credibility that any political-ecological model needs to meet is the one given herein.