A mean-field approach for modeling the propagation of perturbations in biochemical reaction networks

Often, the time evolution of a biochemical reaction network is crucial for determining the effects of combining multiple pharmaceuticals. Here we illustrate a mathematical framework for modeling the dominant temporal behaviour of a complicated molecular pathway or biochemical reaction network in response to an arbitrary perturbation, such as resulting from the administration of a therapeutic agent. The method enables the determination of the temporal evolution of a target protein as the perturbation propagates through its regulatory network. The mathematical approach is particularly useful when the experimental data that is available for characterizing or parameterizing the regulatory network is limited or incomplete. To illustrate the method, we consider the examples of the regulatory networks for the target proteins c-Myc and Chop, which play an important role in venetoclax resistance in acute myeloid leukemia. First we show how the networks that regulate each target protein can be reduced to a mean-field model by identifying the distinct effects that groups of proteins in the regulatory network have on the target protein. Then we show how limited proteinlevel data can be used to further simplify the mean-field model to pinpoint the dominant effects of the network perturbation on the target protein. This enables a further reduction in the number of parameters in the model. The result is an ordinary differential equation model that captures the temporal evolution of the expression of a target protein when one or more proteins in its regulatory network have been perturbed. Finally, we show how the dominant effects predicted by the mathematical model agree with RNA sequencing data for the regulatory proteins comprising the molecular network, despite the model not having a priori knowledge of this data. Thus, while the approach gives a simplified model for the expression of the target protein, it allows for the interpretation of the effects of the perturbation on the regulatory network itself. This method can be easily extended to sets of target proteins to model components of a larger systems biology model, and provides an approach for partially integrating RNA sequencing data and protein expression data. Moreover, it is a general approach that can be used to study drug effects on specific protein(s) in any disease or condition.

Metabolic Pathway Analysis of Nitrogen and Phosphorus Uptake by the Consortium between C. vulgaris and P. aeruginosa

Anthropogenic activities have increased the amount of urban wastewater discharged into natural aquatic reservoirs containing a high amount of nutrients such as phosphorus (Pi and PO 4 − 3 ), nitrogen (NH 3 and NO 3 − ) and organic contaminants. Most of the urban wastewater in Mexico do not receive any treatment to remove nutrients. Several studies have reported that an alternative to reduce those contaminants is using consortiums of microalgae and endogenous bacteria. In this research, a genome-scale biochemical reaction network is reconstructed for the co-culture between the microalga Chlorella vulgaris and the bacterium Pseudomonas aeruginosa. Metabolic Pathway Analysis (MPA), is applied to understand the metabolic capabilities of the co-culture and to elucidate the best conditions in removing nutrients. Theoretical yields for phosphorus removal under photoheterotrophic conditions are calculated, determining their values as 0.042 mmol of PO 4 − 3 per g DW of C. vulgaris, 19.43 mmol of phosphorus (Pi) per g DW of C. vulgaris and 4.90 mmol of phosphorus (Pi) per g DW of P. aeruginosa. Similarly, according to the genome-scale biochemical reaction network the theoretical yields for nitrogen removal are 10.3 mmol of NH 3 per g DW of P. aeruginosa and 7.19 mmol of NO 3 − per g DW of C. vulgaris. Thus, this research proves the metabolic capacity of these microorganisms in removing nutrients and their theoretical yields are calculated.

Metabolic Pathway Analysis for Nutrient Removal of the Consortium between C. vulgaris and P. aeruginosa

Anthropogenic activities have increased the amount of urban wastewater discharged into natural aquatic reservoirs confining in them a high amount of nutrients and organics contaminants. Several studies have reported that an alternative to reduce those contaminants is using consortiums of microalgae and endogenous bacteria. In this research, a genome-scale biochemical reaction network is reconstructed for the co-culture between the microalga Chlorella vulgaris and the bacterium Pesudomonas aeruginosa. Metabolic Pathway Analysis (MPA), is applied to understand the metabolic capabilities of the co-culture and to elucidate the best conditions in removing nutrients such as Phosphorus (inorganic phosphorous and phosphate) and Nitrogen (nitrates and ammonia). Theoretical yields for Phosphorus removal under photoheterotrophic conditions are calculated, determining their values as 0.042 mmol of PO4/ g DW of C. vulgaris, 19.53 mmol of inorganic Phosphorus /g DW of C. vulgaris and 4.90 mmol of inorganic Phosphorus/ g DW of P. aeruginosa. Similarly, according to the genome-scale biochemical reaction network the theoretical yields for Nitrogen removal are 10.3 mmol of NH3/g DW of P. aeruginosa and 7.19 mmol of NO3 /g DW of C. vulgaris. Thus, this research proves the metabolic capacity of these microorganisms in removing nutrients and their theoretical yields are calculated.

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab ® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

How spatial heterogeneity shapes multiscale biochemical reaction network dynamics

Spatial heterogeneity in cells can be modelled using distinct compartments connected by molecular movement between them. In addition to movement, changes in the amount of molecules are due to biochemical reactions within compartments, often such that some molecular types fluctuate on a slower timescale than others. It is natural to ask the following questions: how sensitive is the dynamics of molecular types to their own spatial distribution, and how sensitive are they to the distribution of others? What conditions lead to effective homogeneity in biochemical dynamics despite heterogeneity in molecular distribution? What kind of spatial distribution is optimal from the point of view of some downstream product? Within a spatially heterogeneous multiscale model, we consider two notions of dynamical homogeneity (full homogeneity and homogeneity for the fast subsystem), and consider their implications under different timescales for the motility of molecules between compartments. We derive rigorous results for their dynamics and long-term behaviour, and illustrate them with examples of a shared pathway, Michaelis–Menten enzymatic kinetics and autoregulating feedbacks. Using stochastic averaging of fast fluctuations to their quasi-steady-state distribution, we obtain simple analytic results that significantly reduce the complexity and expedite simulation of stochastic compartment models of chemical reactions.