Biological oscillations are periodic changes in various signaling processes crucial for the proper functioning of living organisms. These oscillations are modeled by ordinary differential equations, with coefficient variations leading to diverse periodic behaviors, typically measured by oscillatory frequencies. This paper explores sampling techniques for neural networks to model the relationship between system coefficients and oscillatory frequency. However, the scarcity of oscillations in the vast coefficient space results in many samples exhibiting non-periodic behaviors, and small coefficient changes near oscillation boundaries can significantly alter oscillatory properties. This leads to non-oscillatory bias and boundary sensitivity, making accurate predictions difficult. While existing importance and uncertainty sampling approaches partially mitigate these challenges, they either fail to resolve the sensitivity problem or result in redundant sampling. To address these limitations, we propose the Hierarchical Gradient-based Genetic Sampling (HGGS) framework, which improves the accuracy of neural network predictions for biological oscillations. The first layer, Gradient-based Filtering, extracts sensitive oscillation boundaries and removes redundant non-oscillatory samples, creating a balanced coarse dataset. The second layer, Multi-grid Genetic Sampling, utilizes residual information to refine these boundaries and explore new high-residual regions, increasing data diversity for model training. Experimental results demonstrate that HGGS outperforms seven comparative sampling methods across four biological systems, highlighting its effectiveness in enhancing sampling and prediction accuracy.

The cell cycle of Caulobacter crescentus involves the polar morphogenesis and an asymmetric cell division driven by precise interactions and regulations of proteins, which makes Caulobacter an ideal model organism for investigating bacterial cell development and differentiation. The abundance of molecular data accumulated on Caulobacter motivates system biologists to analyze the complex regulatory network of cell cycle via quantitative modeling. In this paper, We propose a comprehensive model to accurately characterize the underlying mechanisms of cell cycle regulation based on the study of: a) chromosome replication and methylation; b) interactive pathways of five master regulatory proteins including DnaA, GcrA, CcrM, CtrA, and SciP, as well as novel consideration of their corresponding mRNAs; c) cell cycle-dependent proteolysis of CtrA through hierarchical protease complexes. The temporal dynamics of our simulation results are able to closely replicate an extensive set of experimental observations and capture the main phenotype of seven mutant strains of Caulobacter crescentus. Collectively, the proposed model can be used to predict phenotypes of other mutant cases, especially for nonviable strains which are hard to cultivate and observe. Moreover, the module of cyclic proteolysis is an efficient tool to study the metabolism of proteins with similar mechanisms.

Abstract Background Second messengers, c-di-GMP and (p)ppGpp, are vital regulatory molecules in bacteria, influencing cellular processes such as biofilm formation, transcription, virulence, quorum sensing, and proliferation. While c-di-GMP and (p)ppGpp are both synthesized from GTP molecules, they play antagonistic roles in regulating the cell cycle. In C. crescentus, c-di-GMP works as a major regulator of pole morphogenesis and cell development. It inhibits cell motility and promotes S-phase entry by inhibiting the activity of the master regulator, CtrA. Intracellular (p)ppGpp accumulates under starvation, which helps bacteria to survive under stressful conditions through regulating nucleotide levels and halting proliferation. (p)ppGpp responds to nitrogen levels through RelA-SpoT homolog enzymes, detecting glutamine concentration using a nitrogen phosphotransferase system (PTS Ntr). This work relates the guanine nucleotide-based second messenger regulatory network with the bacterial PTS Ntr system and investigates how bacteria respond to nutrient availability. Results We propose a mathematical model for the dynamics of c-di-GMP and (p)ppGpp in C. crescentus and analyze how the guanine nucleotide-based second messenger system responds to certain environmental changes communicated through the PTS Ntr system. Our mathematical model consists of seven ODEs describing the dynamics of nucleotides and PTS Ntr enzymes. Our simulations are consistent with experimental observations and suggest, among other predictions, that SpoT can effectively decrease c-di-GMP levels in response to nitrogen starvation just as well as it increases (p)ppGpp levels. Thus, the activity of SpoT (or its homologues in other bacterial species) can likely influence the cell cycle by influencing both c-di-GMP and (p)ppGpp. Conclusions In this work, we integrate current knowledge and experimental observations from the literature to formulate a novel mathematical model. We analyze the model and demonstrate how the PTS Ntr system influences (p)ppGpp, c-di-GMP, GMP and GTP concentrations. While this model does not consider all aspects of PTS Ntr signaling, such as cross-talk with the carbon PTS system, here we present our first effort to develop a model of nutrient signaling in C. crescentus.

BACKGROUND: Cell size is a key characteristic that significantly affects many aspects of cellular physiology. There are specific control mechanisms during cell cycle that maintain the cell size within a range from generation to generation. Such control mechanisms introduce substantial variabilities to important properties of the cell cycle such as growth and division. To quantitatively study the effect of such variability in progression through cell cycle, detailed stochastic models are required. RESULTS: In this paper, a new hybrid stochastic model is proposed to study the effect of molecular noise and size control mechanism on the variabilities in cell cycle of the budding yeast Saccharomyces cerevisiae. The proposed model provides an accurate, yet computationally efficient approach for simulation of an intricate system by integrating the deterministic and stochastic simulation schemes. The developed hybrid stochastic model can successfully capture several key features of the cell cycle observed in experimental data. In particular, the proposed model: 1) confirms that the majority of noise in size control stems from low copy numbers of transcripts in the G1 phase, 2) identifies the size and time regulation modules in the size control mechanism, and 3) conforms with phenotypes of early G1 mutants in exquisite detail. CONCLUSIONS: Hybrid stochastic modeling approach can be used to provide quantitative descriptions for stochastic properties of the cell cycle within a computationally efficient framework.

BACKGROUND: The hybrid stochastic simulation algorithm, proposed by Haseltine and Rawlings (HR), is a combination of differential equations for traditional deterministic models and Gillespie's algorithm (SSA) for stochastic models. The HR hybrid method can significantly improve the efficiency of stochastic simulations for multiscale biochemical networks. Previous studies on the accuracy analysis for a linear chain reaction system showed that the HR hybrid method is accurate if the scale difference between fast and slow reactions is above a certain threshold, regardless of population scales. However, the population of some reactant species might be driven negative if they are involved in both deterministic and stochastic systems. RESULTS: This work investigates the negativity problem of the HR hybrid method, analyzes and tests it with several models including a linear chain system, a nonlinear reaction system, and a realistic biological cell cycle system. As a benchmark, the second slow reaction firing time is used to measure the effect of negative populations on the accuracy of the HR hybrid method. Our analysis demonstrates that usually the error caused by negative populations is negligible compared with approximation errors of the HR hybrid method itself, and sometimes negativity phenomena may even improve the accuracy. But for systems where negative species are involved in nonlinear reactions or some species are highly sensitive to negative species, the system stability will be influenced and may lead to system failure when using the HR hybrid method. In those circumstances, three remedies are studied for the negativity problem. CONCLUSION: The results of different models and examples suggest that the Zero-Reaction rule is a good remedy for nonlinear and sensitive systems considering its efficiency and simplicity.

Parameter estimation in discrete or continuous deterministic cell cycle models is challenging for several reasons, including the nature of what can be observed, and the accuracy and quantity of those observations. The challenge is even greater for stochastic models, where the number of simulations and amount of empirical data must be even larger to obtain statistically valid parameter estimates. The two main contributions of this work are (1) stochastic model parameter estimation based on directly matching multivariate probability distributions, and (2) a new quasi-Newton algorithm class QNSTOP for stochastic optimization problems. QNSTOP directly uses the random objective function value samples rather than creating ensemble statistics. QNSTOP is used here to directly match empirical and simulated joint probability distributions rather than matching summary statistics. Results are given for a current state-of-the-art stochastic cell cycle model of budding yeast, whose predictions match well some summary statistics and one-dimensional distributions from empirical data, but do not match well the empirical joint distributions. The nature of the mismatch provides insight into the weakness in the stochastic model.

Compartmental models have been used to model information diffusion on social media. However, there have been few studies on modelling positive and negative public opinions using compartmental models. This study aimed for using sentiment analysis and compartmental model to model the propagation of positive and negative opinions on microblogging big media. The authors studied the news propagation of seven popular social topics on China's Sina Weibo microblogging platform. Natural language processing and sentiment analysis were used to identify public opinions from microblogging big data. Then two existing (SIZ and SEIZ) models and a newly developed (SE2IZ) model were implemented to model the news propagation and evaluate the trends of public opinions on selected social topics. Simulation study was used to check model fitting performance. The results show that the new SE2IZ model has a better model fitting performance than existing models. This study sheds some new light on using social media for public opinion estimation and prediction.

Stochastic simulation of reaction-diffusion systems presents great challenges for spatiotemporal biological models. A straightforward extension of Gillespie's stochastic simulation algorithm (SSA) for spatially inhomogeneous systems leads to the Inhomogeneous Stochastic Simulation Algorithm (ISSA). However, the ISSA can be prohibitively expensive in computation, especially when the mesh size is too small. In this paper, a simple formula of the optimal mesh size is developed in the one dimensional case for reaction-diffusion systems. Based on our formula, we proposed a multigrid discretization method for stochastic simulation of multiscale reaction-diffusion systems. With the combination of proper discretization for species in different reaction and diffusion scales, we can greatly reduce the size of the system and achieve high efficiency. Numerical experiment with the stochastic simulation of PopZ polarization and polymerization in Caulobacter cell cycle is presented to demonstrate the accuracy and efficiency of the proposed discretization method.

Insulin secreted by pancreatic islet β-cells is the principal regulating hormone of glucose metabolism. It plays a key role in controlling glucose level in blood. Impairment of the pancreatic islet function may cause glucose to accumulate in the blood, and result in diabetes mellitus. In order to study the cause of dysfunction of pancreatic islets, a multiple-cell model containing healthy and unhealthy cells is proposed based on an existing single cell model. The β-cells in the model are connected through direct electrical connections between neighboring β-cells. The simulation results show that around 20% unhealthy cells in pancreatic islets will disrupt the insulin secretion. This suggests that a small portion of unhealthy cells may have greater effect in the dysfunction of insulin oscillation than expected.

Insulin secreted by pancreatic islet β-cells is the principal regulating hormone of glucose metabolism. Disruption of insulin secretion may cause glucose to accumulate in the blood, and result in diabetes mellitus. Although deterministic models of the insulin secretion pathway are available, the stochastic aspect of the biological pathway has not been explored. As a first step in this direction, we present a hybrid model of the insulin secretion pathway, in which the delayed rectifying K+channels are treated as stochastic events. Simulation results of our hybrid model demonstrate that our model not only can reproduce the bursts of electrical activity as the deterministic model does, but also can be used to predict the magnitude of the total number of the delayed rectifying K+channels per cell needed in order to prevent the function of this pathway from disruption by stochastic effects. The coupling effect of multiple cells is also studied based on the hybrid model, which shows the synchronization behavior of the cells.

The chemical master equation, which is often considered as an accurate stochastic description of general chemical systems, usually imposes intensive computational requirements when used to characterize molecular biological systems. The major challenge comes from the curse of dimensionally, which has been tackled by a few research papers. The essential goal is to aggregate the system efficiently with limited approximation error. This paper presents an adaptive way to implement the aggregation process using information collected from Monte Carlo methods. Numerical results show the effectiveness of the proposed algorithm despite the lack of explicit estimation of approximation error.

We present an approach to reconstructing chemical reaction networks from time series measurements of the concentrations of the molecules involved. Our solution strategy combines techniques from numerical sensitivity analysis and probabilistic graphical models. By modeling a chemical reaction system as a Markov network (undirected graphical model), we show how systematically probing for sensitivities between molecular species can identify the topology of the network. Given the topology, our approach next uses detailed sensitivity profiles to characterize properties of reactions such as reversibility, enzyme-catalysis, and the precise stoichiometries of the reactants and products. We demonstrate applications to reconstructing key biological systems including the yeast cell cycle. In addition to network reconstruction, our algorithm finds applications in model reduction and model comprehension. We argue that our reconstruction algorithm can serve as an important primitive for data mining in systems biology applications.